Question

Find the area of an equilateral triangle with 12-inch altitudes

Answer 1

`\textit{height of an equilateral triangle}\\\\ h=\cfrac{s√(3)}{2}~~ \begin{cases} s=\stackrel{side's}{length}\\[-0.5em] \hrulefill\\ h=12 \end{cases}\implies 12=\cfrac{s√(3)}{2}\implies 24=s√(3)\implies \cfrac{24}{√(3)}=s \\\\[-0.35em] ~\dotfill`

`\textit{area of an equilateral triangle}\\\\ A=\cfrac{s^2√(3)}{4}\qquad \qquad A=\cfrac{~~ \left( (24)/(√(3)) \right)^2 √(3)~~}{2}\implies A=\cfrac{~~ (24^2)/(3) √(3)~~}{2} \\\\\\ A=\cfrac{192√(3)}{2}\implies A=96√(3)\implies A\approx 166.28`