# The altitude of an equilateral triangle is 12. What is the length of a side and what is the area of the triangle?

Dec 9, 2015

Length of one side is $8 \sqrt{3}$ and area is $48 \sqrt{3}$.

#### Explanation:

Let side length, altitude (height), and area be s, h, and A respectively.

$\textcolor{w h i t e}{\times} h = \sqrt{3} \frac{s}{2}$

$\implies s \cdot \frac{\sqrt{3}}{2} \textcolor{red}{\cdot \frac{2}{\sqrt{3}}} = 12 \textcolor{red}{\cdot \frac{2}{\sqrt{3}}}$
$\implies s = 12 \cdot \frac{2}{\sqrt{3}} \textcolor{b l u e}{\cdot \frac{\sqrt{3}}{\sqrt{3}}}$
$\textcolor{w h i t e}{\times x} = 8 \sqrt{3}$

$\textcolor{w h i t e}{\times} A = a \frac{h}{2}$
$\textcolor{w h i t e}{\times x} = 8 \sqrt{3} \cdot \frac{12}{2}$
$\textcolor{w h i t e}{\times x} = 48 \sqrt{3}$