{"id":11405,"date":"2022-07-08T22:24:38","date_gmt":"2022-07-08T16:54:38","guid":{"rendered":"https:\/\/mathemerize.com\/?p=11405"},"modified":"2022-07-08T22:24:43","modified_gmt":"2022-07-08T16:54:43","slug":"diagonals-of-trapezium-abcd-with-ab-dc-intersect-each-other-at-the-point-o-if-ab-2cd-find-the-ratio-of-the-areas-of-triangles-aob-and-cod","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/diagonals-of-trapezium-abcd-with-ab-dc-intersect-each-other-at-the-point-o-if-ab-2cd-find-the-ratio-of-the-areas-of-triangles-aob-and-cod\/","title":{"rendered":"Diagonals of trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangles AOB and COD."},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>In triangles AOB and COD,<img decoding=\"async\" loading=\"lazy\" class=\" wp-image-11425 alignright\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-2.jpeg?resize=241%2C130&#038;ssl=1\" alt=\"triangles\" width=\"241\" height=\"130\" srcset=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-2.jpeg?resize=300%2C162&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-2.jpeg?resize=1024%2C553&amp;ssl=1 1024w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-2.jpeg?resize=768%2C415&amp;ssl=1 768w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-2.jpeg?resize=1536%2C829&amp;ssl=1 1536w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-2.jpeg?w=1700&amp;ssl=1 1700w\" sizes=\"(max-width: 241px) 100vw, 241px\" data-recalc-dims=\"1\" \/><\/p>\n<p>\\(\\angle\\) AOB = \\(\\angle\\) COD\u00a0 \u00a0 \u00a0 \u00a0(vertically opposite angles)<\/p>\n<p>\\(\\angle\\) OAB = \\(\\angle\\) OCD\u00a0 \u00a0 \u00a0 \u00a0 (corresponding angles)<\/p>\n<p>\\(\\therefore\\) By AA similarity,<\/p>\n<p>\\(\\triangle\\) AOB ~ \\(\\triangle\\) COD<\/p>\n<p>\\(\\implies\\)\u00a0 \\(area(\\triangle AOB)\\over area(\\triangle COD)\\) = \\({AB}^2\\over {DC}^2\\)<\/p>\n<p>\\(\\implies\\)\u00a0 \\(area(\\triangle AOB)\\over area(\\triangle COD)\\) = = \\({2DC}^2\\over {DC}^2\\) = \\(4\\over 1\\)<\/p>\n<p>Hence, Area of triangle AOB : Area of triangle COD = <strong>4 : 1<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : In triangles AOB and COD, \\(\\angle\\) AOB = \\(\\angle\\) COD\u00a0 \u00a0 \u00a0 \u00a0(vertically opposite angles) \\(\\angle\\) OAB = \\(\\angle\\) OCD\u00a0 \u00a0 \u00a0 \u00a0 (corresponding angles) \\(\\therefore\\) By AA similarity, \\(\\triangle\\) AOB ~ \\(\\triangle\\) COD \\(\\implies\\)\u00a0 \\(area(\\triangle AOB)\\over area(\\triangle COD)\\) = \\({AB}^2\\over {DC}^2\\) \\(\\implies\\)\u00a0 \\(area(\\triangle AOB)\\over area(\\triangle COD)\\) = = \\({2DC}^2\\over {DC}^2\\) = \\(4\\over &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/diagonals-of-trapezium-abcd-with-ab-dc-intersect-each-other-at-the-point-o-if-ab-2cd-find-the-ratio-of-the-areas-of-triangles-aob-and-cod\/\"> <span class=\"screen-reader-text\">Diagonals of trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangles AOB and COD.<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,912],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Diagonals of trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangles AOB and COD. - Mathemerize<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/diagonals-of-trapezium-abcd-with-ab-dc-intersect-each-other-at-the-point-o-if-ab-2cd-find-the-ratio-of-the-areas-of-triangles-aob-and-cod\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Diagonals of trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangles AOB and COD. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : In triangles AOB and COD, (angle) AOB = (angle) COD\u00a0 \u00a0 \u00a0 \u00a0(vertically opposite angles) (angle) OAB = (angle) OCD\u00a0 \u00a0 \u00a0 \u00a0 (corresponding angles) (therefore) By AA similarity, (triangle) AOB ~ (triangle) COD (implies)\u00a0 (area(triangle AOB)over area(triangle COD)) = ({AB}^2over {DC}^2) (implies)\u00a0 (area(triangle AOB)over area(triangle COD)) = = ({2DC}^2over {DC}^2) = (4over &hellip; Diagonals of trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangles AOB and COD. Read More &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/diagonals-of-trapezium-abcd-with-ab-dc-intersect-each-other-at-the-point-o-if-ab-2cd-find-the-ratio-of-the-areas-of-triangles-aob-and-cod\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2022-07-08T16:54:38+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-07-08T16:54:43+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-2-300x162.jpeg\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/diagonals-of-trapezium-abcd-with-ab-dc-intersect-each-other-at-the-point-o-if-ab-2cd-find-the-ratio-of-the-areas-of-triangles-aob-and-cod\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/diagonals-of-trapezium-abcd-with-ab-dc-intersect-each-other-at-the-point-o-if-ab-2cd-find-the-ratio-of-the-areas-of-triangles-aob-and-cod\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Diagonals of trapezium ABCD with AB || DC intersect each other at the point O. 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If AB = 2CD, find the ratio of the areas of triangles AOB and COD. - Mathemerize","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathemerize.com\/diagonals-of-trapezium-abcd-with-ab-dc-intersect-each-other-at-the-point-o-if-ab-2cd-find-the-ratio-of-the-areas-of-triangles-aob-and-cod\/","og_locale":"en_US","og_type":"article","og_title":"Diagonals of trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangles AOB and COD. - Mathemerize","og_description":"Solution : In triangles AOB and COD, (angle) AOB = (angle) COD\u00a0 \u00a0 \u00a0 \u00a0(vertically opposite angles) (angle) OAB = (angle) OCD\u00a0 \u00a0 \u00a0 \u00a0 (corresponding angles) (therefore) By AA similarity, (triangle) AOB ~ (triangle) COD (implies)\u00a0 (area(triangle AOB)over area(triangle COD)) = ({AB}^2over {DC}^2) (implies)\u00a0 (area(triangle AOB)over area(triangle COD)) = = ({2DC}^2over {DC}^2) = (4over &hellip; Diagonals of trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangles AOB and COD. 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