Question

select the correct answer.

what is the height, \\(x\\), of the equilateral triangle?

8 in.

\\(8\sqrt{3}\\) in.

4 in.

\\(4\sqrt{3}\\) in.

Explanation:

Identify the properties of the triangle

An equilateral triangle has three equal sides of length \(s = 8\text{ in.}\) and three \(60^\circ\) angles.

Relate the height to a special right triangle

The height \(x\) bisects the top angle into two \(30^\circ\) angles and bisects the base into two segments of length:

$$ \frac{s}{2} = \frac{8}{2} = 4\text{ in.} $$

This forms a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) right triangle where:

• The shorter leg is \(4\text{ in.}\)
• The hypotenuse is \(8\text{ in.}\)
• The longer leg (height \(x\)) is opposite the \(60^\circ\) angle.

Calculate the height \(x\)

Using the ratios of a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle:

$$ x = 4\sqrt{3}\text{ in.} $$

Answer:

• (A) \(8\text{ in.}\)
• (B) \(8\sqrt{3}\text{ in.}\)
• (C) \(4\text{ in.}\)
• (D) \(4\sqrt{3}\text{ in.}\) (Correct answer)