Question

A square and an equilateral triangle have equal perimeters. the area of the triangle is $2\sqrt {3}$ square inches. what is the number of inches in the length of the diagonal of the square?3

Answer 1

The question is
A square and an equilateral triangle have equal perimeters. The area of the triangle is 2√3 square inches. What is the number of inches in the length of the diagonal of the square?

we know that
the area of an equilateral triangle is
applying the law of sines
A=(1/2)*b²*sin 60°-----> 2√3=(1/2)*b²*√3/2
(2√3)*(2/√3)=(1/2)*b²
4=(1/2)*b²
b²=8
b=√8 in

perimeter of the triangle=3*b-----> 3*√8 in

let
x----> the length side of the square
perimeter of the square=perimeter of the triangle
perimeter of the square=3*√8 in
and
perimeter of the square=4*x
4*x=3*√8
x=(3/4)*√8----> x=(3/2)*√2 in

find the diagonal of the square
applying the Pythagoras theorem
D²=x²+x²----> D²=2*x²----> D²=2*((3/2)*√2)²
D²=2*(9/4)*2
D²=9
D=3 in

the answer is
the number of inches in the length of the diagonal of the square is 3