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)
Responses
16 inches
16 inches
8 inches
8 inches
43–√ inches
4 Start Root 3 End Root inches
4 inches
Responses
16 inches
16 inches
8 inches
8 inches
43–√ inches
4 Start Root 3 End Root inches
4 inches
Answers
GPT-4o mini
To solve this problem, we first need to determine the length of a side of the equilateral triangle. Given that the perimeter of the equilateral triangle is 24 inches, and since all sides are equal, we can calculate the length of one side as follows:
\[
\text{Length of one side} = \frac{\text{Perimeter}}{3} = \frac{24 \text{ inches}}{3} = 8 \text{ inches}
\]
Now, when we consider the equilateral triangle, we can draw a height from one vertex to the opposite side, thereby creating two 30-60-90 triangles. In a 30-60-90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where:
- The shortest side (which is opposite the \(30^\circ\) angle) is half the length of the hypotenuse.
- The side opposite the \(60^\circ\) angle is \(\sqrt{3}\) times the shortest side.
In our case, the hypotenuse of the 30-60-90 triangle is equal to the length of the side of the equilateral triangle, which is 8 inches. Thus, we can find the length of the side opposite the \(30^\circ\) angle:
\[
\text{Length opposite } 30^\circ = \frac{8 \text{ inches}}{2} = 4 \text{ inches}
\]
Thus, the length of the side that is opposite the \(30^\circ\) angle is **4 inches**. Therefore, the correct response is:
**4 inches**.
\[
\text{Length of one side} = \frac{\text{Perimeter}}{3} = \frac{24 \text{ inches}}{3} = 8 \text{ inches}
\]
Now, when we consider the equilateral triangle, we can draw a height from one vertex to the opposite side, thereby creating two 30-60-90 triangles. In a 30-60-90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where:
- The shortest side (which is opposite the \(30^\circ\) angle) is half the length of the hypotenuse.
- The side opposite the \(60^\circ\) angle is \(\sqrt{3}\) times the shortest side.
In our case, the hypotenuse of the 30-60-90 triangle is equal to the length of the side of the equilateral triangle, which is 8 inches. Thus, we can find the length of the side opposite the \(30^\circ\) angle:
\[
\text{Length opposite } 30^\circ = \frac{8 \text{ inches}}{2} = 4 \text{ inches}
\]
Thus, the length of the side that is opposite the \(30^\circ\) angle is **4 inches**. Therefore, the correct response is:
**4 inches**.