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

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Example Question #2 : Equilateral Triangles

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

Possible Answers:

$\frac{3\sqrt{3}}{2}$ $\displaystyle \frac{3\sqrt{3}}{2}$

$\frac{\sqrt{3}}{2}$ $\displaystyle \frac{\sqrt{3}}{2}$

$\displaystyle 9$

$6\sqrt{3}$ $\displaystyle 6\sqrt{3}$

$\displaystyle 6$

Correct answer:

$\displaystyle 6$

Explanation:

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

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

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

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

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

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

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Example Question #3 : Equilateral Triangles

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

Possible Answers:

$\displaystyle 8$

$4\sqrt{6}$ $\displaystyle 4\sqrt{6}$

$\frac{4\sqrt{6}}{3}$ $\displaystyle \frac{4\sqrt{6}}{3}$

$\frac{8}{3}$ $\displaystyle \frac{8}{3}$

$\frac{8\sqrt{6}}{3}$ $\displaystyle \frac{8\sqrt{6}}{3}$

Correct answer:

$\frac{8\sqrt{6}}{3}$ $\displaystyle \frac{8\sqrt{6}}{3}$

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 $\sqrt{3}$ $\displaystyle \sqrt{3}$ and the side of equilateral triangle which is opposite $\displaystyle 90$ is $\displaystyle 2$ .

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

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

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

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

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

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Example Question #2 : Equilateral Triangles

Example Question #3 : Equilateral Triangles

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