# Math, Geometry

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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?

1. Steve

the triangle of side s has altitude s√3/2
So, it has area 1/2 (s)(s√3/2) = s^2√3/4

So, if s^2√3/4 = 2√3,
s = √8

So, the triangle has perimeter 3√8

If the square has perimeter 3√8, it has side 3√8/4 = 3/2 √2

The diagonal of a square has length s√2, so in this case, the diagonal is 3.

2. Reiny

Let the side of the equilateral triangle be x

area of equilateral triangle = (1/2)x^2 sin60°
= (1/2)x^2 (√3/2) = (√3/4)x^2

(√3/2)x^2 = 2/√3
3x^2 = 4
x = 2/√3

perimeter of triangle = 3x = 6/√3
which is equal to the perimeter of the square, so each side of the square is 6/(4√3) = 3/(2√3)

let the diagonal be d
d^2 = (3/(2√3) )^2 + (3/(2√3) )^2
= 9/12 +9/12 = 18/12 = 6/4
d = √6/√4 = √6/2

3. Reiny

(√3/2)x^2 = 2/√3
3x^2 = 4
x^2 = 8/3
x = √8/√3

whereas Steve took it as 2√3

4. Steve

That's what happens when TEX gets mixed in with text!

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