Question

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

Explanation:

Step1: Divide the equilateral triangle

An equilateral triangle with side - length \(a = 8\) units. When we draw the altitude \(h\) of an equilateral triangle, it divides the equilateral triangle into two congruent right - triangles. The hypotenuse of each right - triangle is the side of the equilateral triangle (\(a = 8\) units), and the base of each right - triangle is \(\frac{a}{2}\) (since the altitude of an equilateral triangle bisects the base). So, \(b=\frac{8}{2}=4\) units.

Step2: Apply the Pythagorean theorem

By the Pythagorean theorem \(h=\sqrt{a^{2}-b^{2}}\). Substitute \(a = 8\) and \(b = 4\) into the formula:

$$
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Another way: For an equilateral triangle with side - length \(a\), the formula for the altitude \(h=\frac{\sqrt{3}}{2}a\). Substitute \(a = 8\) into the formula, \(h=\frac{\sqrt{3}}{2}\times8=4\sqrt{3}\) units.

Answer:

\(4\sqrt{3}\) units