Question

finding the length of an altitude
the sides of an equilateral triangle are 8 units long. what is the length of the altitude of the triangle?
16√5 units
5√2 units
10√2 units
4√3 units

Explanation:

Step1: Recall properties of equilateral triangle

In an equilateral triangle, the altitude splits it into two 30 - 60 - 90 right triangles. The side of the equilateral triangle is the hypotenuse, half of the side is the shorter leg, and the altitude is the longer leg. If the side length of the equilateral triangle is \(a = 8\) units, then the shorter leg (half of the side) is \(\frac{a}{2}=\frac{8}{2} = 4\) units.

Step2: Apply Pythagorean theorem or 30 - 60 - 90 triangle ratios

For a 30 - 60 - 90 triangle, the ratio of the sides is \(1:\sqrt{3}:2\) (shorter leg : longer leg : hypotenuse). Here, the shorter leg is 4 units, so the longer leg (altitude \(h\)) is \(4\sqrt{3}\) units. We can also use the Pythagorean theorem: \(h^{2}+4^{2}=8^{2}\), so \(h^{2}=64 - 16=48\), and \(h = \sqrt{48}=4\sqrt{3}\) units.

Answer:

\(4\sqrt{3}\) units (the last option: \(4\sqrt{3}\) units)