# What is the perimeter of an equilateral triangle if the length of an altitude is 5/sqrt3?

Mar 13, 2016

Perimeter is 10

$\textcolor{red}{\text{Using ratios is a very powerful tool!}}$

#### Explanation:

Let the height of the standardises triangle by $h$
Let the side length of the triangle in the question be $x$

The by ratio of side lengths we have:

$\textcolor{b l u e}{\left(\text{height of target triangle")/("height of standard triangle")= ("side of target triangle")/("side of standard triangle}\right)}$

$\frac{\frac{5}{\sqrt{3}}}{h} = \frac{x}{2}$

$\frac{5}{\sqrt{3}} \times \frac{1}{h} = \frac{x}{2}$

But $h = \sqrt{3}$ giving

$\frac{5}{\sqrt{3}} \times \frac{1}{\sqrt{3}} = \frac{x}{2}$

$x = \frac{2 \times 5}{3}$

But this is the length for only one side. There are three sides so:

Perimeter = $\frac{3 \times 2 \times 5}{3} = \frac{3}{3} \times 2 \times 5 = 10$