Question

If the area of an equilateral triangle is equal to the area of an isosceles triangle whose base and equal sides are 60 cm and 10cm respectively then the side of the equilateral triangle is

Answer 1

Sol:  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 to the square of the ratio of their corresponding sides i .e.But, we have DB = √2AB     {But diagonal of square is √2 times 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.