{"id":11511,"date":"2022-07-12T23:13:37","date_gmt":"2022-07-12T17:43:37","guid":{"rendered":"https:\/\/mathemerize.com\/?p=11511"},"modified":"2022-07-12T23:13:41","modified_gmt":"2022-07-12T17:43:41","slug":"in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/","title":{"rendered":"In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes."},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>Let ABC be and equilateral triangle and let AD \\(\\perp\\) BC.<img decoding=\"async\" loading=\"lazy\" class=\" wp-image-11528 alignright\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-5-16.jpeg?resize=193%2C179&#038;ssl=1\" alt=\"triangle\" width=\"193\" height=\"179\" srcset=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-5-16.jpeg?resize=300%2C278&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-5-16.jpeg?resize=1024%2C949&amp;ssl=1 1024w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-5-16.jpeg?resize=768%2C712&amp;ssl=1 768w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-5-16.jpeg?resize=1536%2C1424&amp;ssl=1 1536w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-5-16.jpeg?w=1700&amp;ssl=1 1700w\" sizes=\"(max-width: 193px) 100vw, 193px\" data-recalc-dims=\"1\" \/><\/p>\n<p>In \\(\\triangle\\) ADB and ADC, we have :<\/p>\n<p>AB = AC\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (given)<\/p>\n<p>AD = AD\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common side of triangle)<\/p>\n<p>and \\(\\angle\\) ADB = \\(\\angle\\) ADB\u00a0 \u00a0 \u00a0 \u00a0 (each 90)<\/p>\n<p>By RHS criteria of congruence, we have :<\/p>\n<p>\\(\\triangle\\)\u00a0 ADB \\(\\cong\\) \\(\\triangle\\) ADC<\/p>\n<p>So, BD = DC or\u00a0 BD = DC = \\(1\\over 2\\) BC\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;..(1)<\/p>\n<p>Since \\(\\triangle\\)\u00a0 ADB is a right triangle, angled at D, by Pythagoras theorem, we have :<\/p>\n<p>\\({AB}^2\\) = \\({AD}^2\\) + \\({BD}^2\\)<\/p>\n<p>\\({AB}^2\\) = \\({AD}^2\\) + \\(({1\\over 2}BC)^2\\)\u00a0 \u00a0 \u00a0 \u00a0 (from 1)<\/p>\n<p>\\({AB}^2\\) = \\({AD}^2\\) + \\({1\\over 4}{BC}^2\\)<\/p>\n<p>\\({AB}^2\\) = \\({AD}^2\\) + \\({AB}^2\\over 4\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (\\(\\therefore\\)\u00a0 BC = AB)<\/p>\n<p>\\({3\\over 4}{AB}^2\\) = \\({AD}^2\\)\u00a0 or\u00a0 <strong>\\(3{AB}^2\\) = \\(4{AD}^2\\)<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : Let ABC be and equilateral triangle and let AD \\(\\perp\\) BC. In \\(\\triangle\\) ADB and ADC, we have : AB = AC\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (given) AD = AD\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common side of triangle) and \\(\\angle\\) ADB = \\(\\angle\\) ADB\u00a0 \u00a0 \u00a0 \u00a0 (each 90) By RHS criteria of &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/\"> <span class=\"screen-reader-text\">In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes.<\/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>In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes. - 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\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : Let ABC be and equilateral triangle and let AD (perp) BC. In (triangle) ADB and ADC, we have : AB = AC\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (given) AD = AD\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common side of triangle) and (angle) ADB = (angle) ADB\u00a0 \u00a0 \u00a0 \u00a0 (each 90) By RHS criteria of &hellip; In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes. Read More &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2022-07-12T17:43:37+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-07-12T17:43:41+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-5-16-300x278.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\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes.\",\"datePublished\":\"2022-07-12T17:43:37+00:00\",\"dateModified\":\"2022-07-12T17:43:41+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/\"},\"wordCount\":118,\"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\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/\",\"url\":\"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/\",\"name\":\"In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes. - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2022-07-12T17:43:37+00:00\",\"dateModified\":\"2022-07-12T17:43:41+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes.\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes. - 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\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/","og_locale":"en_US","og_type":"article","og_title":"In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes. - Mathemerize","og_description":"Solution : Let ABC be and equilateral triangle and let AD (perp) BC. In (triangle) ADB and ADC, we have : AB = AC\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (given) AD = AD\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common side of triangle) and (angle) ADB = (angle) ADB\u00a0 \u00a0 \u00a0 \u00a0 (each 90) By RHS criteria of &hellip; In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes. Read More &raquo;","og_url":"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/","og_site_name":"Mathemerize","article_published_time":"2022-07-12T17:43:37+00:00","article_modified_time":"2022-07-12T17:43:41+00:00","og_image":[{"url":"https:\/\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-5-16-300x278.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\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes.","datePublished":"2022-07-12T17:43:37+00:00","dateModified":"2022-07-12T17:43:41+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/"},"wordCount":118,"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\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/","url":"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/","name":"In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes. - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2022-07-12T17:43:37+00:00","dateModified":"2022-07-12T17:43:41+00:00","breadcrumb":{"@id":"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/in-an-equilateral-triangle-prove-that-three-times-the-square-of-one-side-is-equal-to-four-times-the-square-of-one-its-altitudes\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes."}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - Study Math Online","publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathemerize.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mathemerize.com\/#organization","name":"Mathemerize","url":"https:\/\/mathemerize.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","contentUrl":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","width":140,"height":96,"caption":"Mathemerize"},"image":{"@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.instagram.com\/mathemerize\/"]},{"@type":"Person","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df","name":"mathemerize","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","caption":"mathemerize"},"sameAs":["https:\/\/mathemerize.com"],"url":"https:\/\/mathemerize.com\/author\/mathemerize\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/11511"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=11511"}],"version-history":[{"count":2,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/11511\/revisions"}],"predecessor-version":[{"id":11529,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/11511\/revisions\/11529"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=11511"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=11511"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=11511"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}