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 43–√ inches 4 square root of 3inches 16 inches 16 inches 8 inches 8 inches 4 inches
Answers
GPT-4o mini
To solve this, we first need to understand the relationship between the equilateral triangle and the 30-60-90 triangle formed when we draw a height from one vertex to the midpoint of the opposite side.
1. **Finding the side length of the equilateral triangle:**
The perimeter of the equilateral triangle is given as 24 inches. Since all sides of an equilateral triangle are equal, we can find the length of one side (s) as follows:
\[
\text{Perimeter} = 3s
\]
Setting this equal to 24:
\[
3s = 24 \implies s = \frac{24}{3} = 8 \text{ inches}
\]
2. **Finding the 30-60-90 triangle:**
When we draw the height of the equilateral triangle, it divides it 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, respectively.
- The side length opposite the 30° angle is half the length of the equilateral triangle's side (which is the side length s we just calculated). So, this length is:
\[
\text{Side opposite 30°} = \frac{s}{2} = \frac{8}{2} = 4 \text{ inches}
\]
Thus, the length of the side that is opposite the 30-degree angle of the 30-60-90 triangle is **4 inches**.
1. **Finding the side length of the equilateral triangle:**
The perimeter of the equilateral triangle is given as 24 inches. Since all sides of an equilateral triangle are equal, we can find the length of one side (s) as follows:
\[
\text{Perimeter} = 3s
\]
Setting this equal to 24:
\[
3s = 24 \implies s = \frac{24}{3} = 8 \text{ inches}
\]
2. **Finding the 30-60-90 triangle:**
When we draw the height of the equilateral triangle, it divides it 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, respectively.
- The side length opposite the 30° angle is half the length of the equilateral triangle's side (which is the side length s we just calculated). So, this length is:
\[
\text{Side opposite 30°} = \frac{s}{2} = \frac{8}{2} = 4 \text{ inches}
\]
Thus, the length of the side that is opposite the 30-degree angle of the 30-60-90 triangle is **4 inches**.