GRE Math : How to find the length of the side of an equilateral triangle

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #2 : Equilateral Triangles

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

Possible Answers:

\(\displaystyle \frac{3\sqrt{3}}{2}\)

\(\displaystyle \frac{\sqrt{3}}{2}\)

\(\displaystyle 9\)

\(\displaystyle 6\sqrt{3}\)

\(\displaystyle 6\)

Correct answer:

\(\displaystyle 6\)

Explanation:

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

So let's set-up an equation to solve for \(\displaystyle s\)

\(\displaystyle \frac{s^2\sqrt{3}}{4}=9\sqrt{3}\) Cross multiply.

\(\displaystyle s^2\sqrt{3}=36\sqrt{3}\) 

The \(\displaystyle \sqrt{3}\) cancels out and we get \(\displaystyle s^2=36\).

Then take square root on both sides and we get \(\displaystyle 6\) as the final answer.

Example Question #3 : Equilateral Triangles

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

 

Possible Answers:

\(\displaystyle 8\)

\(\displaystyle 4\sqrt{6}\)

\(\displaystyle \frac{4\sqrt{6}}{3}\)

\(\displaystyle \frac{8}{3}\)

\(\displaystyle \frac{8\sqrt{6}}{3}\)

Correct answer:

\(\displaystyle \frac{8\sqrt{6}}{3}\)

Explanation:

By having a height in an equilateral triangle, the angle is bisected therefore creating two \(\displaystyle 30-60-90\) triangles.

The height is opposite the angle \(\displaystyle 60\). We can set-up a proportion.

Side opposite \(\displaystyle 60\) is \(\displaystyle \sqrt{3}\) and the side of equilateral triangle which is opposite \(\displaystyle 90\) is \(\displaystyle 2\).

\(\displaystyle \frac{4\sqrt{2}}{\sqrt{3}}=\frac{s}{2}\) Cross multiply.

\(\displaystyle 8\sqrt{2}=s\sqrt{3}\) Divide both sides by \(\displaystyle \sqrt{3}\)

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

\(\displaystyle \frac{8\sqrt{6}}{3}=s\)

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