Question

select the correct answer.
what is the height, x, of the equilateral triangle?

14 in.
a. \\( 7\sqrt{3} \\) inches
b. 7 inches
c. \\( 14\sqrt{3} \\) inches
d. 14 inches
e. \\( 7\sqrt{2} \\) inches

Explanation:

Step1: Split triangle into 2 right triangles

When we draw the height \(x\) of the equilateral triangle, it splits the base into two equal parts. The base length is 14 in, so each half is \(\frac{14}{2}=7\) in.

Step2: Apply Pythagorean theorem

The side length of the equilateral triangle is 14 in (all sides equal). For the right triangle, hypotenuse = 14 in, one leg = 7 in, the other leg is \(x\).
The Pythagorean theorem is \(a^2 + b^2 = c^2\), rearranged to \(x = \sqrt{c^2 - a^2}\)

$$ x = \sqrt{14^2 - 7^2} $$

Step3: Calculate the value

First compute the squares: \(14^2=196\), \(7^2=49\)

$$ x = \sqrt{196 - 49} = \sqrt{147} = \sqrt{49 \times 3} = 7\sqrt{3} $$

Answer:

A. \(7\sqrt{3}\) inches