17. Prove that the area of an equilateral triangle described on one side of a square is equal to haltthe area of the equilateral triangle described on one of its diagonals. y

17. Prove that the area of an equilateral triangle described on one si

1.	17. Prove that the area of an equilateral triangle described on one side of a square is equal to haltthe area of the equilateral triangle described on one of its diagonals. y
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 / 1Proof: If two equilateral triangles are similar then all angles are = 60 degrees.Therefore, by AAA similarity criterion,△DBF~△AEBAr(ΔDBF) / Ar(ΔAEB) = DB^2/ AB^2 --------------------(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 getAr(ΔDBF) / Ar(ΔAEB) = (√2AB)^2/ AB^2 = 2 AB^2/ AB^2= 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.