Question

If the perimeter of an equilateral triangle is 24 inches, what is the length of the side that is opposite the 30-degree angle of the 30-60-90 triangle that is formed by half of the equilateral triangle? (1 point)
O 4V3 inches
• 16 inches
• 8 inches
• 4 inches

Answer 1

To solve this problem, we first need to determine the length of a side of the equilateral triangle.

Since the perimeter of the equilateral triangle is 24 inches, we can find the length of one side as follows:

\[
\text{Length of one side} = \frac{\text{Perimeter}}{3} = \frac{24 \text{ inches}}{3} = 8 \text{ inches}
\]

Next, we need to form a 30-60-90 triangle. When we draw an altitude from one vertex of the equilateral triangle to the opposite side, it divides the equilateral triangle into two 30-60-90 triangles.

In a 30-60-90 triangle, the ratio of the lengths of the sides opposite the angles 30°, 60°, and 90° is 1 : √3 : 2.

Let’s assign the lengths based on the triangle formed:

- The length opposite the 30° angle (half the length of the equilateral triangle side) = \( \frac{8 \text{ inches}}{2} = 4 \text{ inches} \)
- The length opposite the 60° angle = \( 4\sqrt{3} \text{ inches} \) (this is the height of the equilateral triangle)

Since the question asks for the length of the side opposite the 30-degree angle, we find that:

\[
\text{Length opposite the 30° angle} = 4 \text{ inches}
\]

Thus, the answer is:

**4 inches**.