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2. Prove that the area of the equilateral triangle describearea of the equilateral triangles described on its diagonal. |
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Answer» Here ABCD is a square, AEB is an equilateral triangle described on the side of the square and DBF is an equilateral triangle described on diagonal BD of the square. To Prove: Ar(ΔDBF) / Ar(ΔAEB) = 2 / 1 Proof: If two equilateral triangles are similar then all angles are = 60 degrees. Therefore, by AAA similarity criterion,△DBF~△AEB Ar(ΔDBF) / Ar(ΔAEB) = DB2/ AB2 --------------------(i) We know that the ratio of the areas of two similar triangles is equal tothe square of the ratio of their corresponding sides i .e.But, we haveDB=√2AB {But diagonal of square is√2times of its side}-----(ii). Substitute equation (ii) in equation (i), we get Ar(ΔDBF) / Ar(ΔAEB) = (√2AB)2/ AB2 = 2 AB2/ AB2= 2 ∴ Area of equilateral triangle described on one side os square is equal to half the area of the equilateral triangle described on one of its diagonals. Like my answer if you find it useful! |
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