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)

Question

Responses

43–√ inches
4 Start Root 3 End Root inches

8 inches
8 inches

4 inches
4 inches

16 inches

Answer 1

To find the length of the side opposite the 30-degree angle in the 30-60-90 triangle formed by bisecting the equilateral triangle, we start with the information given.

1. The perimeter of the equilateral triangle is 24 inches. Therefore, each side of the equilateral triangle is: \[ \text{Side length} = \frac{24 \text{ inches}}{3} = 8 \text{ inches}. \]

2. When we bisect the equilateral triangle, we create a 30-60-90 triangle. In a 30-60-90 triangle, the lengths of the sides are in the ratio:

   • The side opposite the 30-degree angle (shorter leg) is \(x\),
   • The side opposite the 60-degree angle (longer leg) is \(x\sqrt{3}\), and
   • The hypotenuse (which is the side of the equilateral triangle) is \(2x\).

3. Here, the hypotenuse of the 30-60-90 triangle is equal to the side of the equilateral triangle: \[ 2x = \text{hypotenuse} = 8 \text{ inches}. \]

4. Solving for \(x\): \[ x = \frac{8}{2} = 4 \text{ inches}. \]

Thus, the length of the side opposite the 30-degree angle is 4 inches.

The answer is 4 inches.