Question

A square and an equilateral triangle have equal perimeters. the area of the triangle is 


`16 √(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=(t^2\sqrt3)/(4)`

Substitute:

`(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=√(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`