GMAT Quantitative - Calculating the height of an equilateral triangle

What is the height of an equilateral triangle with sidelength 20?

$10 \sqrt{3}$

$5 \sqrt{3}$

$10 \sqrt{2}$

$\frac{10 \sqrt{2}}{3}$

Explanation

The area of an equilateral triangle with sidelength is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{20^{2}\sqrt{3}}{4} = \frac{400\sqrt{3}}{4} = 100\sqrt{3}$

Using this area for and 20 for in the general triangle formula, we can obtain :

$\frac{1}{2} bh = A$

$\frac{1}{2}\cdot 20 \cdot h =100\sqrt{3}$

$10 \cdot h =100\sqrt{3}$

$10 \cdot h \div 10 =100\sqrt{3}\div 10$

$h=10\sqrt{3}$

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is an equilateral triangle, with a side length of . What is the height of the triangle?

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

$3\sqrt{3}$

$2\sqrt{3}$

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

$\frac{3}{2}$

Explanation

We know the length of the side, therefore we can use the formula for the height in an equilateral triangle:

$h=s\frac{\sqrt{3}}{2}$ , where is the length of a side and the length of the height.

Therefore, the final answer is $\frac{3\sqrt{3}}{2}$ .

Given that an equilateral triangle has side lengths equal to , determine it's height in simplest form.

$3\sqrt3$

$\sqrt{27}$

$\sqrt6$

$\sqrt{28}$

Explanation

To solve, we must use pythagorean's theorem given that we know the hypotenuse is and one side length is $\left (\frac{1}{2} \textup{ of } 6 \right )$ . Therefore:

$c^2=a^2+b^2\Rightarrow 6^2=a^2+3^2\Rightarrow 27=a^2$

$a=\sqrt{27}=\sqrt{3^2*3}=3\sqrt3$

An equilateral triangle has a side length of . What is the height of the triangle?

$\frac{5\sqrt{3}}{2}$

$\frac{2\sqrt{5}}{3}$

$\frac{3\sqrt{2}}{5}$

$5\sqrt{2}$

Explanation

The height of an upright equilateral triangle is the perpendicular distance from the center of its base to its top. We can imagine that this line cuts the equilateral triangle into two congruent right triangles whose height is half the length of the original base and whose hypotenuse is the original side length. In these two congruent triangles, their base, which is the height of the equilateral triangle, is the only unknown side length, so we can use the Pythagorean theorem to solve for it: