{"id":150,"date":"2026-09-02T10:24:05","date_gmt":"2026-09-02T10:24:05","guid":{"rendered":"https:\/\/1stmaster.com\/education\/?p=150"},"modified":"2026-09-02T12:15:07","modified_gmt":"2026-09-02T12:15:07","slug":"triangle-class-10-important-question-pyq","status":"publish","type":"post","link":"https:\/\/1stmaster.com\/education\/triangle-class-10-important-question-pyq\/","title":{"rendered":"Triangle Class 10 Important Question PYQ"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Welcome Back, Today i will provide you top best important question of Triangle Chapter of class 10. Those question are already asked in previous Year of CBSE Board. <\/p>\n\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-1\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In the given figure, <math><semantics><mrow><mi>D<\/mi><mi>E<\/mi><mo>\u2225<\/mo><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">DE\\parallel BC<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>3<\/mn><mn>5<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AD}{DB}=\\frac35<\/annotation><\/semantics><\/math>. If <math><semantics><mrow><mi>A<\/mi><mi>C<\/mi><mo>=<\/mo><mn>4.8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AC=4.8<\/annotation><\/semantics><\/math> cm, find the length of <math><semantics><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AE<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"764\" height=\"503\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle.webp\" alt=\"\" class=\"wp-image-170\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle.webp 764w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle-300x198.webp 300w\" sizes=\"(max-width: 764px) 100vw, 764px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-1\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-2\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>Solution:<\/strong><br>Let <math><semantics><mrow><mi>A<\/mi><mi>E<\/mi><mo>=<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AE=x<\/annotation><\/semantics><\/math> cm.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then<math display=\"block\"><semantics><mrow><mi>E<\/mi><mi>C<\/mi><mo>=<\/mo><mi>A<\/mi><mi>C<\/mi><mo>\u2212<\/mo><mi>A<\/mi><mi>E<\/mi><mo>=<\/mo><mn>4.8<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EC=AC-AE=4.8-x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since <math><semantics><mrow><mi>D<\/mi><mi>E<\/mi><mo>\u2225<\/mo><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">DE\\parallel BC<\/annotation><\/semantics><\/math>, by Thales&#8217; theorem,<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>C<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AD}{DB}=\\frac{AE}{EC}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mfrac><mn>3<\/mn><mn>5<\/mn><\/mfrac><mo>=<\/mo><mfrac><mi>x<\/mi><mrow><mn>4.8<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac35=\\frac{x}{4.8-x}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>4.8<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">3(4.8-x)=5x<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>14.4<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">14.4-3x=5x<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>8<\/mn><mi>x<\/mi><mo>=<\/mo><mn>14.4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">8x=14.4<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>1.8<\/mn><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{x=1.8}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, AE=1.8&nbsp;cm\u200b<\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-2\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-3\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In <math><semantics><mrow><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle ABC<\/annotation><\/semantics><\/math>, <math><semantics><mrow><mi>D<\/mi><mi>E<\/mi><mo>\u2225<\/mo><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">DE\\parallel BC<\/annotation><\/semantics><\/math>, such that<math display=\"block\"><semantics><mrow><mi>A<\/mi><mi>D<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;cm<\/mtext><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>A<\/mi><mi>E<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;cm<\/mtext><mo separator=\"true\">,<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">AD=(4x-3)\\text{ cm},\\quad AE=(8x-7)\\text{ cm},<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>B<\/mi><mi>D<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;cm<\/mtext><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>C<\/mi><mi>E<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;cm<\/mtext><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD=(3x-1)\\text{ cm},\\quad CE=(5x-3)\\text{ cm}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Find the value of <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-medium\"><img decoding=\"async\" width=\"300\" height=\"198\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle-300x198.png\" alt=\"\" class=\"wp-image-173\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle-300x198.png 300w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle.webp 764w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-3\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-4\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">By Thales&#8217; theorem,<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AD}{BD}=\\frac{AE}{CE}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math display=\"block\"><semantics><mrow><mfrac><mrow><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><mrow><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{4x-3}{3x-1} = \\frac{8x-7}{5x-3}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Cross-multiplying,<math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(4x-3)(5x-3)=(3x-1)(8x-7)<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>20<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>27<\/mn><mi>x<\/mi><mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>24<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>29<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">20x^2-27x+9=24x^2-29x+7<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>4<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4x^2-2x-2=0<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2-x-1=0<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(2x+1)(x-1)=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So,<math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><mspace width=\"1em\"><\/mspace><mtext>or<\/mtext><mspace width=\"1em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=1\\quad\\text{or}\\quad x=-\\frac12<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since a length cannot be negative, x=1\u200b<\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-4\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-5\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In the adjoining figure, <math><semantics><mrow><mi>D<\/mi><mi>E<\/mi><mo>\u2225<\/mo><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">DE\\parallel BC<\/annotation><\/semantics><\/math>. If <math><semantics><mrow><mi>A<\/mi><mi>D<\/mi><mo>=<\/mo><mn>1.7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AD=1.7<\/annotation><\/semantics><\/math> cm, <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>=<\/mo><mn>6.8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AB=6.8<\/annotation><\/semantics><\/math> cm and <math><semantics><mrow><mi>A<\/mi><mi>C<\/mi><mo>=<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AC=9<\/annotation><\/semantics><\/math> cm, find <math><semantics><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AE<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"764\" height=\"503\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle.webp\" alt=\"\" class=\"wp-image-173\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle.webp 764w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle-300x198.png 300w\" sizes=\"(max-width: 764px) 100vw, 764px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-5\" hidden>Show More<\/button><\/section>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-6\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By Thales&#8217; theorem,<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AD}{AB}=\\frac{AE}{AC}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mfrac><mn>1.7<\/mn><mn>6.8<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mn>9<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1.7}{6.8}=\\frac{AE}{9}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math display=\"block\"><semantics><mrow><mi>A<\/mi><mi>E<\/mi><mo>=<\/mo><mfrac><mrow><mn>1.7<\/mn><mo>\u00d7<\/mo><mn>9<\/mn><\/mrow><mn>6.8<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">AE=\\frac{1.7\\times9}{6.8}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>A<\/mi><mi>E<\/mi><mo>=<\/mo><mn>2.25<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">AE=2.25<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, AE=2.25&nbsp;cm\u200b<\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-6\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-7\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">If <math><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E<\/annotation><\/semantics><\/math> are points on the sides <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AC<\/annotation><\/semantics><\/math>, respectively, of <math><semantics><mrow><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle ABC<\/annotation><\/semantics><\/math>, such that<math display=\"block\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>=<\/mo><mn>5.6<\/mn><mtext>&nbsp;cm<\/mtext><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>A<\/mi><mi>D<\/mi><mo>=<\/mo><mn>1.4<\/mn><mtext>&nbsp;cm<\/mtext><mo separator=\"true\">,<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">AB=5.6\\text{ cm},\\quad AD=1.4\\text{ cm},<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>A<\/mi><mi>C<\/mi><mo>=<\/mo><mn>7.2<\/mn><mtext>&nbsp;cm<\/mtext><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>A<\/mi><mi>E<\/mi><mo>=<\/mo><mn>1.8<\/mn><mtext>&nbsp;cm<\/mtext><mo separator=\"true\">,<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">AC=7.2\\text{ cm},\\quad AE=1.8\\text{ cm},<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">show that <math><semantics><mrow><mi>D<\/mi><mi>E<\/mi><mo>\u2225<\/mo><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">DE\\parallel BC<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"764\" height=\"503\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle.webp\" alt=\"\" class=\"wp-image-170\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle.webp 764w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle-300x198.webp 300w\" sizes=\"(max-width: 764px) 100vw, 764px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-7\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-8\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>Solution:<\/strong><math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1.4<\/mn><mn>5.6<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AD}{AB} = \\frac{1.4}{5.6} = \\frac14<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1.8<\/mn><mn>7.2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AE}{AC} = \\frac{1.8}{7.2} = \\frac14<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AD}{AB}=\\frac{AE}{AC}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, by the <strong>converse of Thales&#8217; theorem<\/strong>, DE\u2225BC\u200b<\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-8\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-9\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In the given figure,<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>C<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AD}{DB}=\\frac{AE}{EC}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and <math><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mi>D<\/mi><mi>E<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mi>C<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle ADE=\\angle ACB<\/annotation><\/semantics><\/math>. Prove that <math><semantics><mrow><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle ABC<\/annotation><\/semantics><\/math> is an isosceles triangle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"764\" height=\"503\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle.webp\" alt=\"\" class=\"wp-image-170\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle.webp 764w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/image-triangle-300x198.webp 300w\" sizes=\"(max-width: 764px) 100vw, 764px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-9\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-10\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Given,<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>C<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AD}{DB}=\\frac{AE}{EC}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, by the converse of Thales&#8217; theorem,<math display=\"block\"><semantics><mrow><mi>D<\/mi><mi>E<\/mi><mo>\u2225<\/mo><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">DE\\parallel BC<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mi>D<\/mi><mi>E<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle ADE=\\angle ABC<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">because they are corresponding angles.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But given,<math display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mi>D<\/mi><mi>E<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mi>C<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle ADE=\\angle ACB<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore,<math display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>A<\/mi><mi>C<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle ABC=\\angle ACB<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Angles opposite equal sides are equal, so<math display=\"block\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>=<\/mo><mi>A<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB=AC<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, \u25b3ABC&nbsp;is&nbsp;an&nbsp;isosceles&nbsp;triangle\u200b<\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-10\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-11\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>Question:<\/strong><br>In the given figure, <math><semantics><mrow><mi>D<\/mi><mi>E<\/mi><mo>\u2225<\/mo><mi>O<\/mi><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">DE\\parallel OQ<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>D<\/mi><mi>F<\/mi><mo>\u2225<\/mo><mi>O<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">DF\\parallel OR<\/annotation><\/semantics><\/math>. Show that <math><semantics><mrow><mi>E<\/mi><mi>F<\/mi><mo>\u2225<\/mo><mi>Q<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EF\\parallel QR<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" width=\"1024\" height=\"851\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/7-1024x851.png\" alt=\"\" class=\"wp-image-195\" style=\"aspect-ratio:1.2032887751490566;width:249px;height:auto\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/7-1024x851.png 1024w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/7-300x249.png 300w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/7-767x637.png 767w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/7.webp 1376w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-11\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-12\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In \u0394POQ, DE || OQ.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2234<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>P<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>O<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>P<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>Q<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{PD}{DO}=\\frac{PE}{EQ} \\qquad &#8230;(i)\\quad [\\text{by Thales&#8217; theorem}]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In \u0394POR, DF || OR.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2234<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>P<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>O<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>P<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>R<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{PD}{DO}=\\frac{PF}{FR} \\qquad &#8230;(ii)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From (i) and (ii), we get:<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>P<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>Q<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>P<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>R<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mrow><mo fence=\"true\">[<\/mo><mtext>each&nbsp;equal&nbsp;to&nbsp;<\/mtext><mfrac><mrow><mi>P<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>O<\/mi><\/mrow><\/mfrac><mo fence=\"true\">]<\/mo><\/mrow><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{PE}{EQ}=\\frac{PF}{FR} \\qquad \\left[\\text{each equal to }\\frac{PD}{DO}\\right].<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, in \u0394PQR, E and F are points on PQ and PR respectively such that<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>P<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>Q<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>P<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>R<\/mi><\/mrow><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{PE}{EQ}=\\frac{PF}{FR}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, <strong>EF || QR<\/strong> [by the converse of Thales&#8217; theorem].<\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-12\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-13\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In the given figure, <math><semantics><mrow><mi>P<\/mi><mi>Q<\/mi><mo>\u2225<\/mo><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PQ\\parallel AB<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>P<\/mi><mi>R<\/mi><mo>\u2225<\/mo><mi>A<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PR\\parallel AC<\/annotation><\/semantics><\/math>. Prove that <math><semantics><mrow><mi>Q<\/mi><mi>R<\/mi><mo>\u2225<\/mo><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">QR\\parallel BC<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"267\" height=\"253\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/8.webp\" alt=\"\" class=\"wp-image-190\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-13\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-14\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>SOLUTION<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In \u0394OAB, PQ || AB.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2234<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>O<\/mi><mi>P<\/mi><\/mrow><mrow><mi>P<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>O<\/mi><mi>Q<\/mi><\/mrow><mrow><mi>Q<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;the&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{OP}{PA}=\\frac{OQ}{QB} \\qquad &#8230;(i)\\quad [\\text{by the Thales&#8217; theorem}]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In \u0394AOC, PR || AC.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2234<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>O<\/mi><mi>P<\/mi><\/mrow><mrow><mi>P<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>O<\/mi><mi>R<\/mi><\/mrow><mrow><mi>R<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{OP}{PA}=\\frac{OR}{RC} \\qquad &#8230;(ii)\\quad [\\text{by Thales&#8217; theorem}]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From (i) and (ii), we get:<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>O<\/mi><mi>Q<\/mi><\/mrow><mrow><mi>Q<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>O<\/mi><mi>R<\/mi><\/mrow><mrow><mi>R<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mspace width=\"1em\"><\/mspace><mtext>in&nbsp;\u0394OBC<\/mtext><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{OQ}{QB}=\\frac{OR}{RC} \\quad\\text{in \u0394OBC}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, in \u0394OBC, O and R are points on OB and OC respectively such that<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>O<\/mi><mi>Q<\/mi><\/mrow><mrow><mi>Q<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>O<\/mi><mi>R<\/mi><\/mrow><mrow><mi>R<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{OQ}{QB}=\\frac{OR}{RC}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, by the converse of Thales&#8217; theorem, <strong>QR || BC<\/strong>.<\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-14\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-15\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In the given figure, <strong>LM || CB<\/strong> and <strong>LN || CD<\/strong>.<br>Prove that<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>M<\/mi><\/mrow><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>N<\/mi><\/mrow><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AM}{AB}=\\frac{AN}{AD}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"267\" height=\"253\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/9.webp\" alt=\"\" class=\"wp-image-189\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-15\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-16\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>SOLUTION<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In \u0394ALM, LM || CB.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><mrow><mi>A<\/mi><mi>M<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><mrow><mi>A<\/mi><mi>L<\/mi><\/mrow><\/mfrac><mo>\u21d2<\/mo><mfrac><mrow><mi>A<\/mi><mi>M<\/mi><\/mrow><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>L<\/mi><\/mrow><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{AB}{AM}=\\frac{AC}{AL} \\Rightarrow \\frac{AM}{AB}=\\frac{AL}{AC} \\qquad &#8230;(i)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In \u0394ALN, LN || CD.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><mrow><mi>A<\/mi><mi>L<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><mrow><mi>A<\/mi><mi>N<\/mi><\/mrow><\/mfrac><mo>\u21d2<\/mo><mfrac><mrow><mi>A<\/mi><mi>L<\/mi><\/mrow><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>N<\/mi><\/mrow><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{AC}{AL}=\\frac{AD}{AN} \\Rightarrow \\frac{AL}{AC}=\\frac{AN}{AD} \\qquad &#8230;(ii)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From (i) and (ii), we get:<math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>A<\/mi><mi>M<\/mi><\/mrow><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>N<\/mi><\/mrow><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{AM}{AB}=\\frac{AN}{AD}}.<\/annotation><\/semantics><\/math><\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-16\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-17\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In the given figure, <strong>DE || AC<\/strong> and <strong>DF || AE<\/strong>.<br>Prove that<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>E<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BF}{FE}=\\frac{BE}{EC}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"634\" height=\"514\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/10.webp\" alt=\"\" class=\"wp-image-193\" style=\"width:323px;height:auto\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/10.webp 634w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/10-300x243.png 300w\" sizes=\"(max-width: 634px) 100vw, 634px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-17\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-18\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In \u0394BAE, DF || AE.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>E<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{BD}{DA}=\\frac{BF}{FE} \\qquad &#8230;(i)\\quad [\\text{by Thales&#8217; theorem}]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In \u0394BAC, DE || AC.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{BD}{DA}=\\frac{BE}{EC} \\qquad &#8230;(ii)\\quad [\\text{by Thales&#8217; theorem}]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From (i) and (ii), we get:<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>B<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>E<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mrow><mo fence=\"true\">[<\/mo><mtext>each&nbsp;equal&nbsp;to&nbsp;<\/mtext><mfrac><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo fence=\"true\">]<\/mo><\/mrow><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{BF}{FE}=\\frac{BE}{EC} \\qquad \\left[\\text{each equal to }\\frac{BD}{DA}\\right].<\/annotation><\/semantics><\/math><\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-18\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-19\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In the given figure, <strong>AB || DE<\/strong> and <strong>BD || EF<\/strong>.<br>Prove that DC2=CF\u00d7AC.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" width=\"934\" height=\"1024\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/11-934x1024.webp\" alt=\"\" class=\"wp-image-197\" style=\"aspect-ratio:0.9121264652468218;width:220px;height:auto\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/11-934x1024.webp 934w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/11-274x300.webp 274w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/11-768x842.webp 768w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/11.webp 1198w\" sizes=\"(max-width: 934px) 100vw, 934px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-19\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-20\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>SOLUTION<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In \u0394ABC, AB || DE.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>C<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{CD}{DA}=\\frac{CE}{EB} \\qquad &#8230;(i)\\quad [\\text{by Thales&#8217; theorem}]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In \u0394CDB, BD || EF.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>C<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>B<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{CF}{FD}=\\frac{CE}{EB} \\qquad &#8230;(ii)\\quad [\\text{by Thales&#8217; theorem}]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From (i) and (ii), we get:<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>C<\/mi><mi>D<\/mi><\/mrow><mrow><mi>D<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>D<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{CD}{DA}=\\frac{CF}{FD}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>D<\/mi><mi>A<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>F<\/mi><mi>D<\/mi><\/mrow><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>taking&nbsp;reciprocals<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow\\quad \\frac{DA}{DC}=\\frac{FD}{CF} \\qquad [\\text{taking reciprocals}]<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>D<\/mi><mi>A<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mfrac><mrow><mi>F<\/mi><mi>D<\/mi><\/mrow><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow\\quad \\frac{DA}{DC}+1=\\frac{FD}{CF}+1<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>D<\/mi><mi>A<\/mi><mo>+<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>F<\/mi><mi>D<\/mi><mo>+<\/mo><mi>C<\/mi><mi>F<\/mi><\/mrow><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow\\quad \\frac{DA+DC}{DC}=\\frac{FD+CF}{CF}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>D<\/mi><mi>C<\/mi><\/mrow><mrow><mi>C<\/mi><mi>F<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow\\quad \\frac{AC}{DC}=\\frac{DC}{CF}<\/annotation><\/semantics><\/math> \u21d2DC2=CF\u00d7AC\u200b.<\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-20\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-21\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In the given figure <strong>PA, QB and RC each is perpendicular to AC<\/strong> such that<math display=\"block\"><semantics><mrow><mi>P<\/mi><mi>A<\/mi><mo>=<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>R<\/mi><mi>C<\/mi><mo>=<\/mo><mi>y<\/mi><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>Q<\/mi><mi>B<\/mi><mo>=<\/mo><mi>z<\/mi><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>A<\/mi><mi>B<\/mi><mo>=<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PA=x,\\quad RC=y,\\quad QB=z,\\quad AB=a<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and<math display=\"block\"><semantics><mrow><mi>B<\/mi><mi>C<\/mi><mo>=<\/mo><mi>b<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC=b.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Prove that<math display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mi>x<\/mi><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><mi>y<\/mi><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mi>z<\/mi><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac1x+\\frac1y=\\frac1z.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"267\" height=\"253\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/12.webp\" alt=\"\" class=\"wp-image-188\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-21\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-22\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>SOLUTION<\/strong><math display=\"block\"><semantics><mrow><mi>P<\/mi><mi>A<\/mi><mo>\u22a5<\/mo><mi>A<\/mi><mi>C<\/mi><mtext>&nbsp;and&nbsp;<\/mtext><mi>Q<\/mi><mi>B<\/mi><mo>\u22a5<\/mo><mi>A<\/mi><mi>C<\/mi><mo>\u21d2<\/mo><mi>Q<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>P<\/mi><mi>A<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PA\\perp AC\\text{ and }QB\\perp AC \\Rightarrow QB\\parallel PA.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, in \u0394PAC, QB || PA. So, \u0394QBC \u223c \u0394PAC.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>Q<\/mi><mi>B<\/mi><\/mrow><mrow><mi>P<\/mi><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>\u21d2<\/mo><mfrac><mi>z<\/mi><mi>x<\/mi><\/mfrac><mo>=<\/mo><mfrac><mi>b<\/mi><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;the&nbsp;property&nbsp;of&nbsp;similar&nbsp;<\/mtext><mi mathvariant=\"normal\">\u25b3<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{QB}{PA}=\\frac{BC}{AC} \\Rightarrow \\frac{z}{x}=\\frac{b}{a+b} \\qquad &#8230;(i)\\quad [\\text{by the property of similar }\\triangle]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In \u0394RAC, QB || RC. So, \u0394QBC \u223c \u0394RAC.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>Q<\/mi><mi>B<\/mi><\/mrow><mrow><mi>R<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>\u21d2<\/mo><mfrac><mi>z<\/mi><mi>y<\/mi><\/mfrac><mo>=<\/mo><mfrac><mi>a<\/mi><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;the&nbsp;property&nbsp;of&nbsp;similar&nbsp;<\/mtext><mi mathvariant=\"normal\">\u25b3<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{QB}{RC}=\\frac{AB}{AC} \\Rightarrow \\frac{z}{y}=\\frac{a}{a+b} \\qquad &#8230;(ii)\\quad [\\text{by the property of similar }\\triangle]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From (i) and (ii), we get:<math display=\"block\"><semantics><mrow><mfrac><mi>z<\/mi><mi>x<\/mi><\/mfrac><mo>+<\/mo><mfrac><mi>z<\/mi><mi>y<\/mi><\/mfrac><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>b<\/mi><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mi>a<\/mi><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{z}{x}+\\frac{z}{y} = \\left(\\frac{b}{a+b}+\\frac{a}{a+b}\\right)=1<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mi>z<\/mi><mi>x<\/mi><\/mfrac><mo>+<\/mo><mfrac><mi>z<\/mi><mi>y<\/mi><\/mfrac><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow\\quad \\frac{z}{x}+\\frac{z}{y}=1<\/annotation><\/semantics><\/math>you can show the steps clearly by <strong>taking z common<\/strong>:<math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mi>z<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>1<\/mn><mi>x<\/mi><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><mi>y<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow z\\left(\\frac{1}{x}+\\frac{1}{y}\\right)=1<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mfrac><mn>1<\/mn><mi>x<\/mi><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><mi>y<\/mi><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mi>z<\/mi><\/mfrac><mspace width=\"2em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>dividing&nbsp;both&nbsp;sides&nbsp;by&nbsp;<\/mtext><mi>z<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow \\frac{1}{x}+\\frac{1}{y}=\\frac{1}{z} \\qquad [\\text{dividing both sides by }z]<\/annotation><\/semantics><\/math><\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-22\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-23\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">ABCD is a trapezium with <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB\\parallel DC<\/annotation><\/semantics><\/math>.<br>E and F are points on non-parallel sides <math><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math> respectively such that <math><semantics><mrow><mi>E<\/mi><mi>F<\/mi><mo>\u2225<\/mo><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EF\\parallel AB<\/annotation><\/semantics><\/math>. Show that<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AE}{ED}=\\frac{BF}{FC}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"467\" height=\"367\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/17.webp\" alt=\"\" class=\"wp-image-183\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/17.webp 467w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/17-300x236.webp 300w\" sizes=\"(max-width: 467px) 100vw, 467px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-23\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-24\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>SOLUTION<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>GIVEN<\/strong> A trap. <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABCD<\/annotation><\/semantics><\/math> in which <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB\\parallel DC<\/annotation><\/semantics><\/math>. E and F are points on <math><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>B<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BC<\/annotation><\/semantics><\/math> respectively such that <math><semantics><mrow><mi>E<\/mi><mi>F<\/mi><mo>\u2225<\/mo><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EF\\parallel AB<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>TO PROVE<\/strong><math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AE}{ED}=\\frac{BF}{FC}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>CONSTRUCTION<\/strong> Join <math><semantics><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AC<\/annotation><\/semantics><\/math>, intersecting <math><semantics><mrow><mi>E<\/mi><mi>F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EF<\/annotation><\/semantics><\/math> at <math><semantics><mrow><mi>G<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">G<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>PROOF<\/strong> <math><semantics><mrow><mi>E<\/mi><mi>F<\/mi><mo>\u2225<\/mo><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EF\\parallel AB<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><mo>\u21d2<\/mo><mi>E<\/mi><mi>F<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB\\parallel DC\\Rightarrow EF\\parallel DC<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now, in <math><semantics><mrow><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle ADC<\/annotation><\/semantics><\/math>, <math><semantics><mrow><mi>E<\/mi><mi>G<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EG\\parallel DC<\/annotation><\/semantics><\/math>.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>G<\/mi><\/mrow><mrow><mi>G<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{AE}{ED}=\\frac{AG}{GC} \\qquad &#8230;(i)\\quad [\\text{by Thales&#8217; theorem}]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Similarly, in <math><semantics><mrow><mi mathvariant=\"normal\">\u25b3<\/mi><mi>C<\/mi><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle CAB<\/annotation><\/semantics><\/math>, <math><semantics><mrow><mi>G<\/mi><mi>F<\/mi><mo>\u2225<\/mo><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">GF\\parallel AB<\/annotation><\/semantics><\/math>.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>A<\/mi><mi>G<\/mi><\/mrow><mrow><mi>G<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>F<\/mi><\/mrow><mrow><mi>F<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mrow><mo fence=\"true\">[<\/mo><mo>\u2234<\/mo><mfrac><mrow><mi>G<\/mi><mi>C<\/mi><\/mrow><mrow><mi>A<\/mi><mi>G<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>F<\/mi><mi>C<\/mi><\/mrow><mrow><mi>B<\/mi><mi>F<\/mi><\/mrow><\/mfrac><mtext>&nbsp;by&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{AG}{GC}=\\frac{BF}{FC} \\qquad &#8230;(ii)\\quad \\left[\\therefore\\frac{GC}{AG}=\\frac{FC}{BF}\\text{ by Thales&#8217; theorem}\\right]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From (i) and (ii), we get: EDAE\u200b=FCBF\u200b\u200b.<\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-24\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-25\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">ABCD is a trapezium in which <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB\\parallel DC<\/annotation><\/semantics><\/math> and its diagonals intersect each other at the point O.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Prove that<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AO}{OC}=\\frac{BO}{OD}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"467\" height=\"367\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/17.webp\" alt=\"\" class=\"wp-image-186\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/17.webp 467w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/17-300x236.png 300w\" sizes=\"(max-width: 467px) 100vw, 467px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-25\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-26\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>SOLUTION<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>GIVEN<\/strong> A trapezium <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABCD<\/annotation><\/semantics><\/math> in which <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB\\parallel DC<\/annotation><\/semantics><\/math> and its diagonals <math><semantics><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AC<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD<\/annotation><\/semantics><\/math> intersect at O.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>TO PROVE<\/strong><math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AO}{OC}=\\frac{BO}{OD}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>CONSTRUCTION<\/strong> Through O, draw <math><semantics><mrow><mi>E<\/mi><mi>O<\/mi><mo>\u2225<\/mo><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EO\\parallel AB<\/annotation><\/semantics><\/math>, meeting <math><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math> at E.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>PROOF<\/strong> In <math><semantics><mrow><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle ADC<\/annotation><\/semantics><\/math>, <math><semantics><mrow><mi>E<\/mi><mi>O<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EO\\parallel DC<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mo stretchy=\"false\">[<\/mo><mo>\u2234<\/mo><mi>E<\/mi><mi>O<\/mi><mo>\u2225<\/mo><mi>A<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">[\\therefore EO\\parallel AB\\parallel DC]<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{AE}{ED}=\\frac{AO}{OC} \\qquad &#8230;(i)\\quad [\\text{by Thales&#8217; theorem}]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In <math><semantics><mrow><mi mathvariant=\"normal\">\u25b3<\/mi><mi>D<\/mi><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle DAB<\/annotation><\/semantics><\/math>, <math><semantics><mrow><mi>E<\/mi><mi>O<\/mi><mo>\u2225<\/mo><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EO\\parallel AB<\/annotation><\/semantics><\/math>.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{AE}{ED}=\\frac{BO}{OD} \\qquad &#8230;(ii)<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mtext>From&nbsp;(i)&nbsp;and&nbsp;(ii),&nbsp;we&nbsp;get:<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\text{From (i) and (ii), we get:}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mrow><mi>A<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>D<\/mi><\/mrow><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{AO}{OC}=\\frac{BO}{OD}}.<\/annotation><\/semantics><\/math><\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-26\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-27\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">The diagonals of a quadrilateral <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABCD<\/annotation><\/semantics><\/math> intersect each other at the point O such that<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AO}{OC}=\\frac{BO}{OD}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Show that <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABCD<\/annotation><\/semantics><\/math> is a trapezium.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"467\" height=\"367\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/17.webp\" alt=\"\" class=\"wp-image-186\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/17.webp 467w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/17-300x236.png 300w\" sizes=\"(max-width: 467px) 100vw, 467px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-27\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-28\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>SOLUTION<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>GIVEN<\/strong> A quadrilateral <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABCD<\/annotation><\/semantics><\/math> whose diagonals <math><semantics><mrow><mi>A<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AC<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>B<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">BD<\/annotation><\/semantics><\/math> intersect at a point O such that<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AO}{OC}=\\frac{BO}{OD}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>TO PROVE<\/strong> <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABCD<\/annotation><\/semantics><\/math> is a trapezium, i.e., <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB\\parallel DC<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>CONSTRUCTION<\/strong> Draw <math><semantics><mrow><mi>E<\/mi><mi>O<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EO\\parallel DC<\/annotation><\/semantics><\/math>, meeting <math><semantics><mrow><mi>A<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AD<\/annotation><\/semantics><\/math> at E.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>PROOF<\/strong> In <math><semantics><mrow><mi mathvariant=\"normal\">\u25b3<\/mi><mi>A<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\triangle ACD<\/annotation><\/semantics><\/math>, <math><semantics><mrow><mi>E<\/mi><mi>O<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EO\\parallel DC<\/annotation><\/semantics><\/math>.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>A<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mspace width=\"2em\"><\/mspace><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mi mathvariant=\"normal\">.<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">[<\/mo><mtext>by&nbsp;Thales\u2019&nbsp;theorem<\/mtext><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{AO}{OC}=\\frac{AE}{ED} \\qquad &#8230;(i)\\quad [\\text{by Thales&#8217; theorem}]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But,<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mspace width=\"1em\"><\/mspace><mtext>(given)<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AO}{OC}=\\frac{BO}{OD}\\quad\\text{(given)}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mfrac><mrow><mi>B<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>A<\/mi><mi>E<\/mi><\/mrow><mrow><mi>E<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mspace width=\"1em\"><\/mspace><mtext>in&nbsp;<\/mtext><mi mathvariant=\"normal\">\u25b3<\/mi><mi>D<\/mi><mi>A<\/mi><mi>B<\/mi><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad \\frac{BO}{OD}=\\frac{AE}{ED} \\quad\\text{in }\\triangle DAB.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So, <math><semantics><mrow><mi>E<\/mi><mi>O<\/mi><mo>\u2225<\/mo><mi>A<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EO\\parallel AB<\/annotation><\/semantics><\/math> [by the converse of Thales&#8217; theorem].<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But, <math><semantics><mrow><mi>E<\/mi><mi>O<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">EO\\parallel DC<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>A<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{AB\\parallel DC}.<\/annotation><\/semantics><\/math><\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-28\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--question wp-block-qa-collapse-blocks-question\" data-word-limit=\"2500\" data-kind=\"question\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Question<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-29\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\">In the given figure, <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABCD<\/annotation><\/semantics><\/math> is a trapezium in which <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB\\parallel DC<\/annotation><\/semantics><\/math> and its diagonals intersect at O. If<math display=\"block\"><semantics><mrow><mi>A<\/mi><mi>O<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;cm<\/mtext><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>O<\/mi><mi>C<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;cm<\/mtext><mo separator=\"true\">,<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">AO=(3x-1)\\text{ cm},\\quad OC=(5x-3)\\text{ cm},<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>B<\/mi><mi>O<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;cm&nbsp;and&nbsp;<\/mtext><mi>O<\/mi><mi>D<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;cm<\/mtext><mo separator=\"true\">,<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">BO=(2x+1)\\text{ cm and }OD=(6x-5)\\text{ cm},<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">find the value of <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"412\" height=\"317\" src=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/19.webp\" alt=\"\" class=\"wp-image-182\" srcset=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/19.webp 412w, https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/09\/19-300x231.webp 300w\" sizes=\"(max-width: 412px) 100vw, 412px\" \/><\/figure>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-29\" hidden>Show More<\/button><\/section>\n\n<section class=\"qa-collapse-block qa-collapse-block--answer wp-block-qa-collapse-blocks-answer\" data-word-limit=\"1500\" data-kind=\"answer\" data-join-next=\"false\"><div class=\"qa-collapse-block__header\"><span class=\"qa-collapse-block__badge\">Answer<\/span><span class=\"qa-collapse-block__number\" aria-hidden=\"true\"><\/span><\/div><div id=\"qa-collapse-content-30\" class=\"qa-collapse-block__full\">\n\n<p class=\"wp-block-paragraph\"><strong>SOLUTION<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We know that <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mo>\u2225<\/mo><mi>D<\/mi><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">AB\\parallel DC<\/annotation><\/semantics><\/math> in trap. <math><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABCD<\/annotation><\/semantics><\/math> and its diagonals intersect at O. Then, we have:<math display=\"block\"><semantics><mrow><mfrac><mrow><mi>A<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>C<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>B<\/mi><mi>O<\/mi><\/mrow><mrow><mi>O<\/mi><mi>D<\/mi><\/mrow><\/mfrac><mo>\u21d2<\/mo><mfrac><mrow><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mrow><mn>6<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{AO}{OC}=\\frac{BO}{OD} \\Rightarrow \\frac{3x-1}{5x-3}=\\frac{2x+1}{6x-5}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow (3x-1)(6x-5)=(2x+1)(5x-3)<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mn>18<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>21<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo>=<\/mo><mn>10<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow 18x^2-21x+5=10x^2-x-3<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mn>8<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>20<\/mn><mi>x<\/mi><mo>+<\/mo><mn>8<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow 8x^2-20x+8=0<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow 2x^2-5x+2=0<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow (x-2)(2x-1)=0<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mtext>&nbsp;or&nbsp;<\/mtext><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Rightarrow x=2\\text{ or }x=\\frac12.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But, <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac12<\/annotation><\/semantics><\/math> will make<math display=\"block\"><semantics><mrow><mi>O<\/mi><mi>C<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;cm<\/mtext><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mn>5<\/mn><mo>\u00d7<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>\u2212<\/mo><mn>3<\/mn><mo fence=\"true\">)<\/mo><\/mrow><mtext>&nbsp;cm<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">OC=(5x-3)\\text{ cm} =\\left(5\\times\\frac12-3\\right)\\text{ cm}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mtext>&nbsp;cm<\/mtext><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">=-\\frac12\\text{ cm}.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">And, the distance cannot be negative.<math display=\"block\"><semantics><mrow><mo>\u2234<\/mo><mspace width=\"1em\"><\/mspace><mi>x<\/mi><mo>\u2260<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi mathvariant=\"normal\">.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\therefore\\quad x\\ne\\frac12.<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, x=2\u200b.<\/p>\n\n<\/div><button class=\"qa-collapse-block__toggle\" type=\"button\" aria-expanded=\"true\" aria-controls=\"qa-collapse-content-30\" hidden>Show More<\/button><\/section>","protected":false},"excerpt":{"rendered":"<p>Welcome Back, Today i will provide you top best important question of Triangle Chapter of class 10. Those question are [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":166,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[3],"tags":[],"class_list":["post-150","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-10"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.0.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"Welcome Back, Today i will provide you top best important question of Triangle Chapter of class 10. Those question are already asked in previous Year of CBSE Board. In the given figure, DE\u2225BCDE\\parallel BC and ADDB=35\\frac{AD}{DB}=\\frac35. If AC=4.8AC=4.8 cm, find the length of AEAE. Solution:Let AE=xAE=x cm. ThenEC=AC\u2212AE=4.8\u2212xEC=AC-AE=4.8-x Since DE\u2225BCDE\\parallel BC, by Thales&#039; theorem,ADDB=AEEC\\frac{AD}{DB}=\\frac{AE}{EC}35=x4.8\u2212x\\frac35=\\frac{x}{4.8-x}3(4.8\u2212x)=5x3(4.8-x)=5x14.4\u22123x=5x14.4-3x=5x8x=14.48x=14.4x=1.8\\boxed{x=1.8} Therefore,\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"1st Master\"\/>\n\t<link rel=\"canonical\" href=\"https:\/\/1stmaster.com\/education\/triangle-class-10-important-question-pyq\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.0.1\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n\t\t<meta name=\"twitter:title\" content=\"Triangle Class 10 Important Question PYQ\" \/>\n\t\t<meta name=\"twitter:description\" content=\"Welcome Back, Today i will provide you top best important question of Triangle Chapter of class 10. Those question are already asked in previous Year of CBSE Board. In the given figure, DE\u2225BCDE\\parallel BC and ADDB=35\\frac{AD}{DB}=\\frac35. If AC=4.8AC=4.8 cm, find the length of AEAE. Solution:Let AE=xAE=x cm. ThenEC=AC\u2212AE=4.8\u2212xEC=AC-AE=4.8-x Since DE\u2225BCDE\\parallel BC, by Thales&#039; theorem,ADDB=AEEC\\frac{AD}{DB}=\\frac{AE}{EC}35=x4.8\u2212x\\frac35=\\frac{x}{4.8-x}3(4.8\u2212x)=5x3(4.8-x)=5x14.4\u22123x=5x14.4-3x=5x8x=14.48x=14.4x=1.8\\boxed{x=1.8} Therefore,\" \/>\n\t\t<meta name=\"twitter:image\" content=\"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/08\/1st-master-education-logo-1.webp\" \/>\n\t\t<!-- All in One SEO -->\n\n","aioseo_head_json":{"title":"Triangle Class 10 Important Question PYQ","description":"Welcome Back, Today i will provide you top best important question of Triangle Chapter of class 10. Those question are already asked in previous Year of CBSE Board. In the given figure, DE\u2225BCDE\\parallel BC and ADDB=35\\frac{AD}{DB}=\\frac35. If AC=4.8AC=4.8 cm, find the length of AEAE. Solution:Let AE=xAE=x cm. ThenEC=AC\u2212AE=4.8\u2212xEC=AC-AE=4.8-x Since DE\u2225BCDE\\parallel BC, by Thales' theorem,ADDB=AEEC\\frac{AD}{DB}=\\frac{AE}{EC}35=x4.8\u2212x\\frac35=\\frac{x}{4.8-x}3(4.8\u2212x)=5x3(4.8-x)=5x14.4\u22123x=5x14.4-3x=5x8x=14.48x=14.4x=1.8\\boxed{x=1.8} Therefore,","canonical_url":"https:\/\/1stmaster.com\/education\/triangle-class-10-important-question-pyq\/","robots":"max-image-preview:large","keywords":"","webmasterTools":{"miscellaneous":""},"schema":null,"twitter:card":"summary_large_image","twitter:title":"Triangle Class 10 Important Question PYQ","twitter:description":"Welcome Back, Today i will provide you top best important question of Triangle Chapter of class 10. Those question are already asked in previous Year of CBSE Board. In the given figure, DE\u2225BCDE\\parallel BC and ADDB=35\\frac{AD}{DB}=\\frac35. If AC=4.8AC=4.8 cm, find the length of AEAE. Solution:Let AE=xAE=x cm. ThenEC=AC\u2212AE=4.8\u2212xEC=AC-AE=4.8-x Since DE\u2225BCDE\\parallel BC, by Thales' theorem,ADDB=AEEC\\frac{AD}{DB}=\\frac{AE}{EC}35=x4.8\u2212x\\frac35=\\frac{x}{4.8-x}3(4.8\u2212x)=5x3(4.8-x)=5x14.4\u22123x=5x14.4-3x=5x8x=14.48x=14.4x=1.8\\boxed{x=1.8} Therefore,","twitter:image":"https:\/\/1stmaster.com\/education\/wp-content\/uploads\/2026\/08\/1st-master-education-logo-1.webp"},"aioseo_meta_data":{"post_id":"150","title":null,"description":null,"keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"focus_keyword":null,"additional_keywords":null,"truseo_locale":null,"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_custom_url":null,"og_image_custom_fields":null,"og_image_url":null,"og_image_width":null,"og_image_height":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_image_url":null,"twitter_title":null,"twitter_description":null,"schema_type":"default","schema_type_options":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"BlogPosting","isEnabled":true},"graphs":[]},"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","local_seo":null,"limit_modified_date":false,"ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"breadcrumb_settings":null,"seo_analyzer_scan_date":null,"created":"2026-09-02 10:16:05","updated":"2026-09-03 01:58:57"},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/1stmaster.com\/education\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/1stmaster.com\/education\/category\/class-10\/\" title=\"Class 10\">Class 10<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">&raquo;<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tTriangle Class 10 Important Question PYQ\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/1stmaster.com\/education"},{"label":"Class 10","link":"https:\/\/1stmaster.com\/education\/category\/class-10\/"},{"label":"Triangle Class 10 Important Question PYQ","link":"https:\/\/1stmaster.com\/education\/triangle-class-10-important-question-pyq\/"}],"_links":{"self":[{"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/posts\/150","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/comments?post=150"}],"version-history":[{"count":18,"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/posts\/150\/revisions"}],"predecessor-version":[{"id":198,"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/posts\/150\/revisions\/198"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/media\/166"}],"wp:attachment":[{"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/media?parent=150"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/categories?post=150"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/1stmaster.com\/education\/wp-json\/wp\/v2\/tags?post=150"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}