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 4 inches 4 inches 43–√ inches 4 Start Root 3 End Root inches 8 inches
Answers
GPT-4o mini
First, we determine the length of a side of the equilateral triangle. Since the perimeter of the triangle is 24 inches, we can find the length of one side by dividing the perimeter by 3:
\[
\text{Length of one side} = \frac{24 \text{ inches}}{3} = 8 \text{ inches}
\]
Next, we consider the 30-60-90 triangle formed by drawing an altitude from one vertex to the midpoint of the opposite side of the equilateral triangle. This altitude divides the equilateral triangle into two equal halves, creating two 30-60-90 triangles.
In a 30-60-90 triangle, the ratios of the lengths of the sides opposite the 30-degree, 60-degree, and 90-degree angles are \(1 : \sqrt{3} : 2\).
In this case:
- The side opposite the 30-degree angle is half of the side of the equilateral triangle.
- The length of the side opposite the 60-degree angle is the altitude of the triangle.
- The hypotenuse is the side of the equilateral triangle.
Since the length of one side of the equilateral triangle is 8 inches, the length of the side opposite the 30-degree angle, which corresponds to half of the equilateral triangle's side, is:
\[
\text{Side opposite 30 degrees} = \frac{8 \text{ inches}}{2} = 4 \text{ inches}
\]
Therefore, the length of the side that is opposite the 30-degree angle is:
\[
\text{4 inches}
\]
\[
\text{Length of one side} = \frac{24 \text{ inches}}{3} = 8 \text{ inches}
\]
Next, we consider the 30-60-90 triangle formed by drawing an altitude from one vertex to the midpoint of the opposite side of the equilateral triangle. This altitude divides the equilateral triangle into two equal halves, creating two 30-60-90 triangles.
In a 30-60-90 triangle, the ratios of the lengths of the sides opposite the 30-degree, 60-degree, and 90-degree angles are \(1 : \sqrt{3} : 2\).
In this case:
- The side opposite the 30-degree angle is half of the side of the equilateral triangle.
- The length of the side opposite the 60-degree angle is the altitude of the triangle.
- The hypotenuse is the side of the equilateral triangle.
Since the length of one side of the equilateral triangle is 8 inches, the length of the side opposite the 30-degree angle, which corresponds to half of the equilateral triangle's side, is:
\[
\text{Side opposite 30 degrees} = \frac{8 \text{ inches}}{2} = 4 \text{ inches}
\]
Therefore, the length of the side that is opposite the 30-degree angle is:
\[
\text{4 inches}
\]