Question

a square and an equilateral triangle have equal perimeters. the area of the triangle is 
$16 \sqrt{3}$

square centimeters. how long, in centimeters, is a diagonal of the square? express your answer in simplest radical form.

Answer 1

a - length of side of a square

t - length of side of a triangle

The perimeter of a square: $P_S=4a$

The perimeter of a triangle: $P_T=3t$

We have the area of a triangle: $A_T=16\sqrt3\ cm^2$

The formula of an area of an equilateral trinagle: $A_T=\dfrac{t^2\sqrt3}{4}$

Substitute:

$\dfrac{t^2\sqrt3}{4}=16\sqrt3$    multiply both sides by 4

$t^2\sqrt3=64\sqrt3$     divide both sides by $\sqrt3$

$t^2=64\to t=\sqrt{64}\to t=8\ cm$

The perimeter of a triangle: $P_T=3(8)=24\ cm$

Substitute to the formula of a perimeter of a square:

$4a=24$       divide both sides by 4

$a=6\ cm$

The formula of a diagonal of a square: $d=a\sqrt2$

Substitute:

$d=6\sqrt2\ cm$