Question

A square and an equilateral triangle have equal perimeters. The area of the triangle is
16ã3 square centimeters, how long in centimeters is a diagonal of the square? Express your answer in simplest radical form.

Answer 1

let each side of the square be x cm
let each side of the equilateral triangle be 2y cm (I use 2y to avoid fractions)

so 4x = 6y,
x = 3y/2

using Pythagoras, the height of the triangle is (√3)y
so
(1/2)(2y)(√3)y = 16√3
which solves to y = 4

then x = 3(4)/2 = 6
let the diagonal of the square be d
d^2 = 6^2 + 6^2
d = √72
= 6√2