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

Question

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

Savannah Aguilar · Accepted Answer

Length of one side is #8sqrt3# and area is #48sqrt3#.

Let side length, altitude (height), and area be s, h, and A respectively. #color(white)(xx)h=sqrt3s/2# #=>s*sqrt3/2color(red)(*2/sqrt3)=12color(red)(*2/sqrt3)# #=>s=12*2/sqrt3color(blue)(*sqrt3/sqrt3)# #color(white)(xxx)=8sqrt3# #color(white)(xx)A=ah/2# #color(white)(xxx)=8sqrt3*12/2# #color(white)(xxx)=48sqrt3#

Mason Adams · Answer

undefined The length of a side of the equilateral triangle can be found using the formula $ \text{side length} = \frac{2 \times \text{altitude}}{\sqrt{3}} $, where the altitude is given as 12.

Substitute the altitude into the formula:
$$ \text{side length} = \frac{2 \times 12}{\sqrt{3}} $$
$$ \text{side length} = \frac{24}{\sqrt{3}} $$
$$ \text{side length} = \frac{24 \times \sqrt{3}}{3} $$
$$ \text{side length} = 8\sqrt{3} $$

The area of an equilateral triangle can be calculated using the formula $ \text{area} = \frac{\sqrt{3}}{4} \times \text{side length}^2 $.

Substitute the side length into the area formula:
$$ \text{area} = \frac{\sqrt{3}}{4} \times (8\sqrt{3})^2 $$
$$ \text{area} = \frac{\sqrt{3}}{4} \times 192 $$
$$ \text{area} = \frac{192\sqrt{3}}{4} $$
$$ \text{area} = 48\sqrt{3} $$

So, the length of a side of the equilateral triangle is $ 8\sqrt{3} $ units, and the area of the triangle is $ 48\sqrt{3} $ square units.