Given an equilateral triangle with height $15$ units, what is the length of side $x$?

• A. $30$
• B. $15÷\sqrt{3}$
• C. $15\times \sqrt{3}$
• D. $\frac{30}{\sqrt{3}}$

Answer: (C)

Verified step by step guidance

1. Recall that in an equilateral triangle, all sides are equal, and the height splits the triangle into two 30-60-90 right triangles.
2. In a 30-60-90 triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite 30° is half the hypotenuse (which is the side of the equilateral triangle), and the side opposite 60° is the height.
3. Let the side length of the equilateral triangle be \(x\). The height corresponds to the side opposite the 60° angle, which is \(\frac{\sqrt{3}}{2} x\).
4. Set up the equation using the given height: \(15 = \frac{\sqrt{3}}{2} x\).
5. Solve for \(x\) by multiplying both sides by \(\frac{2}{\sqrt{3}}\): \(x = 15 \times \frac{2}{\sqrt{3}}\).