An equilateral triangle has sides of length $s$. What is the length of its altitude?

• A. $\sqrt{s}$
• B. $\frac{s}{2}$
• C. $\frac{\sqrt{3}s}{2}$
• D. $\frac{s}{\sqrt{3}}$

Answer: (C)

Verified step by step guidance

1. Recall that an equilateral triangle has all sides equal and all angles equal to 60 degrees.
2. To find the altitude, draw a perpendicular from one vertex to the opposite side, which will bisect that side into two segments of length \( \frac{s}{2} \).
3. This altitude forms a right triangle with the altitude as one leg, half the side \( \frac{s}{2} \) as the other leg, and the side \( s \) as the hypotenuse.
4. Use the Pythagorean theorem: \( \text{altitude}^2 + \left( \frac{s}{2} \right)^2 = s^2 \).
5. Solve for the altitude: \( \text{altitude} = \sqrt{s^2 - \left( \frac{s}{2} \right)^2} = \frac{\sqrt{3}}{2} s \).