Question

The area of an equilateral triangle is LaTeX: 108\sqrt[]{3}\:ft^2 108 3 f t 2 . What is the length of one side of the triangle?

Answer 1

The length of one side is 12√3.

We know that the area of a triangle is given by the formula:

A=1/2(b)(h)

We can substitute the area in:
108√3 = 1/2(b)(h)

Let b be the side of the equilateral triangle.  To find the height, we will use the Pythagorean theorem.  We know that the height bisects the base, so we will call that leg of the right triangle formed (1/2b).  Since the triangle is equilateral, we will call the hypotenuse b as well.  We now have:

$(\frac{1}{2}b)^2+h^2=b^2 \\ \\\frac{1}{4}b^2+h^2=b^2 \\ \\\frac{1}{4}b^2+h^2-\frac{1}{4}b^2=b^2-\frac{1}{4}b^2 \\ \\h^2=\frac{3}{4}b^2 \\ \\\sqrt{h^2}=\sqrt{\frac{3}{4}b^2} \\ \\h=\frac{\sqrt{3}}{2}b$

We will now substitute this in the formula for area we had above:
108√3=(1/2)(b)(√3/2b)
108√3=√3/4b²

Multiply both sides by 4:
(108√3)×4=(√3/4b²)×4
432√3=√3b²

Divide both sides by √3:
432√3/√3 = √3b²/√3
432=b²

Take the square root of both sides:
√432=√b²

Simplifying the radical, we have 12√3.