{"id":11414,"date":"2022-07-08T22:51:07","date_gmt":"2022-07-08T17:21:07","guid":{"rendered":"https:\/\/mathemerize.com\/?p=11414"},"modified":"2022-07-08T22:51:11","modified_gmt":"2022-07-08T17:21:11","slug":"the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/","title":{"rendered":"The areas of two similar triangles are in the ratio of the square of the corresponding medians."},"content":{"rendered":"<h2>Solution :<\/h2>\n<p><strong>Given<\/strong> : \\(\\triangle\\) ABC ~ \\(\\triangle\\) DEF and AP, DQ are their medians.<img decoding=\"async\" loading=\"lazy\" class=\"wp-image-11435  alignright\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-6-e1657300777294-300x134.jpeg?resize=259%2C116&#038;ssl=1\" alt=\"triangles\" width=\"259\" height=\"116\" srcset=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-6-e1657300777294.jpeg?resize=300%2C134&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-6-e1657300777294.jpeg?resize=1024%2C458&amp;ssl=1 1024w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-6-e1657300777294.jpeg?resize=768%2C344&amp;ssl=1 768w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-6-e1657300777294.jpeg?resize=1536%2C687&amp;ssl=1 1536w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-6-e1657300777294.jpeg?w=1623&amp;ssl=1 1623w\" sizes=\"(max-width: 259px) 100vw, 259px\" data-recalc-dims=\"1\" \/><\/p>\n<p><strong>To Prove<\/strong> : \\(area(\\triangle ABC)\\over area(\\triangle DEF)\\) = \\({AP}^2\\over {DQ}^2\\)<\/p>\n<p><strong>Proof<\/strong> : Since the ratio of the area of two similar triangles is equal to the ratio of the squares of any two corresponding sides.<\/p>\n<p>\\(\\therefore\\)\u00a0 \\(area (\\triangle ABC)\\over area (\\triangle DEF)\\) = \\({AB}^2\\over {DE}^2\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &#8230;&#8230;&#8230;..(1)<\/p>\n<p>Now,\u00a0 \\(\\triangle\\) ABC ~ \\(\\triangle\\) DEF<\/p>\n<p>\\(\\implies\\)\u00a0 \\(AB\\over DE\\) = \\(BC\\over EF\\) = \\(2BP\\over 2EQ\\) = \\(BP\\over EQ\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;&#8230;&#8230;.(2)<\/p>\n<p>Thus, \\(AB\\over DE\\) = \\(BP\\over EQ\\) and \\(\\angle\\) B = \\(\\angle\\) E\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(because \\(\\triangle\\) ABC ~ \\(\\triangle\\) DEF)<\/p>\n<p>By SAS similarity,<\/p>\n<p>\\(\\triangle\\) APB ~ \\(\\triangle\\) DQE<\/p>\n<p>So, \\(BP\\over EQ\\) = \\(AP\\over DQ\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;&#8230;.(3)<\/p>\n<p>From (2) and (3), we get<\/p>\n<p>\\(AB\\over DE\\) = \\(AP\\over DQ\\)\u00a0 \u00a0 \u00a0\\(\\implies\\)\u00a0 \\({AB}^2\\over {DE}^2\\) = \\({AP}^2\\over {DQ}^2\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &#8230;&#8230;&#8230;(4)<\/p>\n<p>Substituting the value of \\({AB}^2\\over {DE}^2\\) from (4) in (1), we get<\/p>\n<p><strong>\\(area (\\triangle ABC)\\over area (\\triangle DEF)\\) = \\({AP}^2\\over {DQ}^2\\)<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : Given : \\(\\triangle\\) ABC ~ \\(\\triangle\\) DEF and AP, DQ are their medians. To Prove : \\(area(\\triangle ABC)\\over area(\\triangle DEF)\\) = \\({AP}^2\\over {DQ}^2\\) Proof : Since the ratio of the area of two similar triangles is equal to the ratio of the squares of any two corresponding sides. \\(\\therefore\\)\u00a0 \\(area (\\triangle ABC)\\over area &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/\"> <span class=\"screen-reader-text\">The areas of two similar triangles are in the ratio of the square of the corresponding medians.<\/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>The areas of two similar triangles are in the ratio of the square of the corresponding medians. - 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\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The areas of two similar triangles are in the ratio of the square of the corresponding medians. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : Given : (triangle) ABC ~ (triangle) DEF and AP, DQ are their medians. To Prove : (area(triangle ABC)over area(triangle DEF)) = ({AP}^2over {DQ}^2) Proof : Since the ratio of the area of two similar triangles is equal to the ratio of the squares of any two corresponding sides. (therefore)\u00a0 (area (triangle ABC)over area &hellip; The areas of two similar triangles are in the ratio of the square of the corresponding medians. 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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\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/","og_locale":"en_US","og_type":"article","og_title":"The areas of two similar triangles are in the ratio of the square of the corresponding medians. - Mathemerize","og_description":"Solution : Given : (triangle) ABC ~ (triangle) DEF and AP, DQ are their medians. To Prove : (area(triangle ABC)over area(triangle DEF)) = ({AP}^2over {DQ}^2) Proof : Since the ratio of the area of two similar triangles is equal to the ratio of the squares of any two corresponding sides. (therefore)\u00a0 (area (triangle ABC)over area &hellip; The areas of two similar triangles are in the ratio of the square of the corresponding medians. Read More &raquo;","og_url":"https:\/\/mathemerize.com\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/","og_site_name":"Mathemerize","article_published_time":"2022-07-08T17:21:07+00:00","article_modified_time":"2022-07-08T17:21:11+00:00","og_image":[{"url":"https:\/\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-6-e1657300777294-300x134.jpeg"}],"author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"The areas of two similar triangles are in the ratio of the square of the corresponding medians.","datePublished":"2022-07-08T17:21:07+00:00","dateModified":"2022-07-08T17:21:11+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/"},"wordCount":163,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"articleSection":["Maths Questions","Similar Triangles Questions"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/","url":"https:\/\/mathemerize.com\/the-areas-of-two-similar-triangles-are-in-the-ratio-of-the-square-of-the-corresponding-medians\/","name":"The areas of two similar triangles are in the ratio of the square of the corresponding medians. - 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