Question

Each side of an equilateral triangle \( ABC \) is \( a \). Prove that the altitude of the triangle is \( \frac{\sqrt{3}}{2} a \).

Hint:

Use Pythagoras’ theorem in an equilateral triangle by splitting it into two right triangles.

Answer 1

In an equilateral triangle, the altitude divides the triangle into two right-angled triangles.
Using Pythagoras’ theorem:
\[ h^2 + \left(\frac{a}{2}\right)^2 = a^2. \] \[ h^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4}. \] \[ h = \frac{\sqrt{3}}{2} a. \]