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 the equilateral triangle?
Answers
GPT-4o mini
To find the length of the side opposite the 30-degree angle in the 30-60-90 triangle formed by half of an equilateral triangle with a perimeter of 24 inches, we can follow these steps:
1. **Calculate the side length of the equilateral triangle**:
The perimeter of an equilateral triangle is given as 24 inches. Since all three sides are equal in length, we can find the length of one side by dividing the perimeter by 3:
\[
\text{Side length} = \frac{\text{Perimeter}}{3} = \frac{24 \text{ inches}}{3} = 8 \text{ inches}
\]
2. **Identify the half-equilateral triangle**:
When we draw a line from one vertex of the equilateral triangle to the midpoint of the opposite side, it splits the triangle into two 30-60-90 triangles.
3. **Determine the side lengths in the 30-60-90 triangle**:
In a 30-60-90 triangle, the ratios of the lengths of the sides are as follows:
- The side opposite the 30-degree angle (let's denote it as \( x \)) is half of the hypotenuse.
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
In our case, the hypotenuse of the 30-60-90 triangle corresponds to the side length of the equilateral triangle, which is \( 8 \) inches.
4. **Calculate the side opposite the 30-degree angle**:
Since the hypotenuse is \( 8 \) inches, the side opposite the 30-degree angle can be calculated as:
\[
x = \frac{8}{2} = 4 \text{ inches}
\]
Thus, the length of the side that is opposite the 30-degree angle of the 30-60-90 triangle is \( \boxed{4} \) inches.
1. **Calculate the side length of the equilateral triangle**:
The perimeter of an equilateral triangle is given as 24 inches. Since all three sides are equal in length, we can find the length of one side by dividing the perimeter by 3:
\[
\text{Side length} = \frac{\text{Perimeter}}{3} = \frac{24 \text{ inches}}{3} = 8 \text{ inches}
\]
2. **Identify the half-equilateral triangle**:
When we draw a line from one vertex of the equilateral triangle to the midpoint of the opposite side, it splits the triangle into two 30-60-90 triangles.
3. **Determine the side lengths in the 30-60-90 triangle**:
In a 30-60-90 triangle, the ratios of the lengths of the sides are as follows:
- The side opposite the 30-degree angle (let's denote it as \( x \)) is half of the hypotenuse.
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
In our case, the hypotenuse of the 30-60-90 triangle corresponds to the side length of the equilateral triangle, which is \( 8 \) inches.
4. **Calculate the side opposite the 30-degree angle**:
Since the hypotenuse is \( 8 \) inches, the side opposite the 30-degree angle can be calculated as:
\[
x = \frac{8}{2} = 4 \text{ inches}
\]
Thus, the length of the side that is opposite the 30-degree angle of the 30-60-90 triangle is \( \boxed{4} \) inches.