a square and an equilateral triangle have equal perimeter.The area of the triangle is 163 square centimeters. How long,centimeters,is a diagonal of the square?

Question

Please i need help with this very hard math question.Please reply as soon as possible.

Answer 1

Let S be the length of a side of the equilateral triangle. Drop an altitude from a vertex to split it into two 30-60-90 right triangles. Each has hypotenuse of length S so the short leg has length S/2 and the long leg (the altitude drawn) is (S√3)/2

So the equilateral triangle has area BH/2 = S[(S√3)/2]/2 = (S²√3)/4 which is given as 163 sq. cm.

 

(S²√3)/4 = 163

 

(S²√3) = 652

 

S² = 652/√3

 

S = √(652/√3)

 

The perimeter is 3S which is also the perimeter of the square. So the side of the square is 3S/4 and the diagonal is

 

D = (√2)(3S/4) = (√2)(3√(652/√3)/4) ≈ 20.58 cm.