Question

An altitude of an equilateral triangle has length `6sqrt3`. What is the perimeter of the triangle?

Answer 1

`36`

Explanation:

The three sides of any equilateral triangle are equal.
The three angles of any equilateral triangle all mesure `60^@`
The altitude bisects the base at right angle `(90^@)`

Given altitude `H=6sqrt3`,
By Pythagorean theorem, we know that
`L^2=H^2+(L/2)^2`
`L^2-(L/2)^2=H^2`
`(3/4)L^2=(6sqrt3)^2=108`
`=> L^2=(108xx4)/3=144, => L=12`
Hence, perimeter `=3L=3xx12=36`

Alternative solution :

`Lxxsin60=H`,
`L=H/sin60=(6sqrt3)/(sqrt3/2)=12`
Hence, perimeter `=3L=3xx12=36`