{"id":11412,"date":"2022-07-08T22:46:43","date_gmt":"2022-07-08T17:16:43","guid":{"rendered":"https:\/\/mathemerize.com\/?p=11412"},"modified":"2022-07-08T22:46:47","modified_gmt":"2022-07-08T17:16:47","slug":"d-e-f-are-the-mid-points-of-the-sides-bc-ca-and-ab-respectively-of-a-triangle-abc-determine-the-ratio-of-the-areas-of-triangle-def-and-triangle-abc","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/d-e-f-are-the-mid-points-of-the-sides-bc-ca-and-ab-respectively-of-a-triangle-abc-determine-the-ratio-of-the-areas-of-triangle-def-and-triangle-abc\/","title":{"rendered":"D, E, F are the mid-points of the sides BC, CA and AB respectively of a \\(\\triangle\\) ABC. Determine the ratio of the areas of \\(\\triangle\\) DEF  and \\(\\triangle\\) ABC."},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>Since D and E are the mid-points of the sides BC and CA respectively of \\(\\triangle\\) ABC.<img decoding=\"async\" loading=\"lazy\" class=\" wp-image-11433 alignright\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-5.jpeg?resize=186%2C211&#038;ssl=1\" alt=\"triangle\" width=\"186\" height=\"211\" srcset=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-5.jpeg?resize=264%2C300&amp;ssl=1 264w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-5.jpeg?resize=902%2C1024&amp;ssl=1 902w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-5.jpeg?resize=768%2C872&amp;ssl=1 768w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-5.jpeg?resize=1352%2C1536&amp;ssl=1 1352w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-4-5.jpeg?w=1700&amp;ssl=1 1700w\" sizes=\"(max-width: 186px) 100vw, 186px\" data-recalc-dims=\"1\" \/><\/p>\n<p>\\(\\therefore\\)\u00a0 DE || BA<\/p>\n<p>\\(\\implies\\)\u00a0 DE || FA\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;&#8230;.(1)<\/p>\n<p>Since D and F are the mid-points of the sides BC and AB respectively of \\(\\triangle\\) ABC. Therefore,<\/p>\n<p>DF || CA\u00a0 \\(\\implies\\)\u00a0 DF || AE\u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;&#8230;..(2)<\/p>\n<p>From (1) and (2), we get AFDE is a parallelogram<\/p>\n<p>Similarly, BDEF is a parallelogram.<\/p>\n<p>In triangle DEF and ABC,<\/p>\n<p>\\(\\angle\\) FDE = \\(\\angle\\) A\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (opposite angles of ||gm AFDE)<\/p>\n<p>\\(\\angle\\) DEF = \\(\\angle\\) B\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (opposite angles of ||gm BDEF)<\/p>\n<p>\\(\\therefore\\)\u00a0 By AA similarity,<\/p>\n<p>\\(\\triangle\\) DEF ~ \\(\\triangle\\) ABC<\/p>\n<p>\\(\\implies\\)\u00a0 \\(area(\\triangle DEF)\\over area(\\triangle ABC)\\) = \\({DE}^2\\over {AB}^2\\) = \\(1\\over 4\\)<\/p>\n<p>(The areas of two similar triangles are in the ratio of the squares of the corresponding sides)<\/p>\n<p>Hence, Area of triangle DEF : Area of triangle ABC = <strong>1 : 4<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : Since D and E are the mid-points of the sides BC and CA respectively of \\(\\triangle\\) ABC. \\(\\therefore\\)\u00a0 DE || BA \\(\\implies\\)\u00a0 DE || FA\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;&#8230;.(1) Since D and F are the mid-points of the sides BC and AB respectively of \\(\\triangle\\) ABC. Therefore, DF || CA\u00a0 \\(\\implies\\)\u00a0 DF &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/d-e-f-are-the-mid-points-of-the-sides-bc-ca-and-ab-respectively-of-a-triangle-abc-determine-the-ratio-of-the-areas-of-triangle-def-and-triangle-abc\/\"> <span class=\"screen-reader-text\">D, E, F are the mid-points of the sides BC, CA and AB respectively of a \\(\\triangle\\) ABC. Determine the ratio of the areas of \\(\\triangle\\) DEF  and \\(\\triangle\\) ABC.<\/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>D, E, F are the mid-points of the sides BC, CA and AB respectively of a \\(\\triangle\\) ABC. Determine the ratio of the areas of \\(\\triangle\\) DEF and \\(\\triangle\\) ABC. - 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\/d-e-f-are-the-mid-points-of-the-sides-bc-ca-and-ab-respectively-of-a-triangle-abc-determine-the-ratio-of-the-areas-of-triangle-def-and-triangle-abc\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"D, E, F are the mid-points of the sides BC, CA and AB respectively of a \\(\\triangle\\) ABC. Determine the ratio of the areas of \\(\\triangle\\) DEF and \\(\\triangle\\) ABC. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : Since D and E are the mid-points of the sides BC and CA respectively of (triangle) ABC. (therefore)\u00a0 DE || BA (implies)\u00a0 DE || FA\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;&#8230;.(1) Since D and F are the mid-points of the sides BC and AB respectively of (triangle) ABC. Therefore, DF || CA\u00a0 (implies)\u00a0 DF &hellip; D, E, F are the mid-points of the sides BC, CA and AB respectively of a (triangle) ABC. Determine the ratio of the areas of (triangle) DEF and (triangle) ABC. 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Determine the ratio of the areas of \\(\\triangle\\) DEF and \\(\\triangle\\) ABC. - 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\/d-e-f-are-the-mid-points-of-the-sides-bc-ca-and-ab-respectively-of-a-triangle-abc-determine-the-ratio-of-the-areas-of-triangle-def-and-triangle-abc\/","og_locale":"en_US","og_type":"article","og_title":"D, E, F are the mid-points of the sides BC, CA and AB respectively of a \\(\\triangle\\) ABC. Determine the ratio of the areas of \\(\\triangle\\) DEF and \\(\\triangle\\) ABC. - Mathemerize","og_description":"Solution : Since D and E are the mid-points of the sides BC and CA respectively of (triangle) ABC. 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