How to find the length of the side of an equilateral triangle - GRE Quantitative Reasoning

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Question

What is the length of a side of an equilateral triangle if the area is $9\sqrt{3}$ ?

Answer

The area of an equilateral triangle is $\frac{s^2\sqrt{3}}{4}$ .

So let's set-up an equation to solve for .

$\frac{s^2\sqrt{3}}{4}=9\sqrt{3}$ Cross multiply.

$s^2\sqrt{3}=36\sqrt{3}$

The $\sqrt{3}$ cancels out and we get .

Then take square root on both sides and we get as the final answer.

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Question

If the height of the equilateral triangle is $4\sqrt{2}$ , then what is the length of a side of an equilateral triangle?

Answer

By having a height in an equilateral triangle, the angle is bisected therefore creating two triangles.

The height is opposite the angle . We can set-up a proportion.

Side opposite is $\sqrt{3}$ and the side of equilateral triangle which is opposite is .

$\frac{4\sqrt{2}}{\sqrt{3}}=\frac{s}{2}$ Cross multiply.

$8\sqrt{2}=s\sqrt{3}$ Divide both sides by $\sqrt{3}$

$\frac{8\sqrt{2}}{\sqrt{3}}=s$ Multiply top and bottom by $\sqrt{3}$ to get rid of the radical.

$\frac{8\sqrt{6}}{3}=s$

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