Posted by **?????????** on Friday, August 2, 2013 at 9:22pm.

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?

Please give me a full clear explanation with the answer. Thanks!!!!!!!

- Math, Geometry -
**Steve**, Friday, August 2, 2013 at 11:49pm
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.

- Math, Geometry -
**Reiny**, Friday, August 2, 2013 at 11:53pm
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

- arithmetic errror - go with Steve Math, Geometry -
**Reiny**, Saturday, August 3, 2013 at 12:01am
made a silly error ...

(√3/2)x^2 = 2/√3

3x^2 = **4**

**x^2 = 8/3
**

x = √8/√3

I also read your area of the triangle as 2/√3

whereas Steve took it as 2√3

- perils of typesetting :-) -
**Steve**, Saturday, August 3, 2013 at 12:04am
That's what happens when T_{E}X gets mixed in with text!

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