{"id":11416,"date":"2022-07-08T22:53:51","date_gmt":"2022-07-08T17:23:51","guid":{"rendered":"https:\/\/mathemerize.com\/?p=11416"},"modified":"2022-07-08T22:53:55","modified_gmt":"2022-07-08T17:23:55","slug":"prove-that-the-area-of-an-equilateral-triangle-described-on-one-side-of-square-is-equal-to-half-the-area-of-the-equilateral-described-on-one-of-the-diagonals","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/prove-that-the-area-of-an-equilateral-triangle-described-on-one-side-of-square-is-equal-to-half-the-area-of-the-equilateral-described-on-one-of-the-diagonals\/","title":{"rendered":"Prove that the area of an equilateral triangle described on one side of square is equal to half the area of the equilateral described on one of the diagonals."},"content":{"rendered":"
Given<\/strong> : A square ABCD, equilateral triangles BCE and ACF have been drawn on side BC and the diagonal AC respectively.<\/p>\n To Prove<\/strong> : \\(area(\\triangle BCE)\\) = \\(1\\over 2\\)\\(area(\\triangle DEF)\\)<\/p>\n Proof<\/strong> : Since all equilateral triangles are similar.<\/p>\n \\(\\implies\\)\u00a0 \\(\\triangle\\) BCE ~ \\(\\triangle\\) ACF<\/p>\n \\(area(\\triangle BCE)\\over area(\\triangle ACF)\\) = \\({BC}^2\\over {AC}^2\\)<\/p>\n Since, Diagonal = \\(\\sqrt{2}\\) side, So, AC = \\(\\sqrt{2}\\) BC<\/p>\n \\(\\implies\\) \\(area(\\triangle BCE)\\over area(\\triangle ACF)\\) = \\({BC}^2\\over {\\sqrt{2} BC}^2\\)<\/p>\n \\(\\implies\\)\u00a0 \\(area(\\triangle BCE)\\over area(\\triangle ACF)\\) = \\(1\\over 2\\)<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":" Solution : Given : A square ABCD, equilateral triangles BCE and ACF have been drawn on side BC and the diagonal AC respectively. To Prove : \\(area(\\triangle BCE)\\) = \\(1\\over 2\\)\\(area(\\triangle DEF)\\) Proof : Since all equilateral triangles are similar. \\(\\implies\\)\u00a0 \\(\\triangle\\) BCE ~ \\(\\triangle\\) ACF \\(area(\\triangle BCE)\\over area(\\triangle ACF)\\) = \\({BC}^2\\over {AC}^2\\) Since, Diagonal …<\/p>\n