{"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":"

Solution :<\/h2>\n

Let ABC be and equilateral triangle and let AD \\(\\perp\\) BC.\"triangle\"<\/p>\n

In \\(\\triangle\\) ADB and ADC, we have :<\/p>\n

AB = AC\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (given)<\/p>\n

AD = AD\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common side of triangle)<\/p>\n

and \\(\\angle\\) ADB = \\(\\angle\\) ADB\u00a0 \u00a0 \u00a0 \u00a0 (each 90)<\/p>\n

By RHS criteria of congruence, we have :<\/p>\n

\\(\\triangle\\)\u00a0 ADB \\(\\cong\\) \\(\\triangle\\) ADC<\/p>\n

So, BD = DC or\u00a0 BD = DC = \\(1\\over 2\\) BC\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0……..(1)<\/p>\n

Since \\(\\triangle\\)\u00a0 ADB is a right triangle, angled at D, by Pythagoras theorem, we have :<\/p>\n

\\({AB}^2\\) = \\({AD}^2\\) + \\({BD}^2\\)<\/p>\n

\\({AB}^2\\) = \\({AD}^2\\) + \\(({1\\over 2}BC)^2\\)\u00a0 \u00a0 \u00a0 \u00a0 (from 1)<\/p>\n

\\({AB}^2\\) = \\({AD}^2\\) + \\({1\\over 4}{BC}^2\\)<\/p>\n

\\({AB}^2\\) = \\({AD}^2\\) + \\({AB}^2\\over 4\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (\\(\\therefore\\)\u00a0 BC = AB)<\/p>\n

\\({3\\over 4}{AB}^2\\) = \\({AD}^2\\)\u00a0 or\u00a0 \\(3{AB}^2\\) = \\(4{AD}^2\\)<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

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 …<\/p>\n

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 »<\/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":"\nIn 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 … In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one its altitudes. 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