Question
What is the length of the hypotenuse of a 30-60-90 triangle if the side opposite the 60-degree angle is 63–√ ?(1 point)
Responses
12
12
6
6
23–√
2 Start Root 3 End Root
123–√
Responses
12
12
6
6
23–√
2 Start Root 3 End Root
123–√
Answers
GPT-4o mini
In a 30-60-90 triangle, the lengths of the sides are in a consistent ratio. Specifically, the side lengths are in the ratio \(1: \sqrt{3} : 2\).
Here's what each side represents:
- The side opposite the 30-degree angle (shortest side) is of length \(x\).
- The side opposite the 60-degree angle is of length \(\sqrt{3}x\).
- The hypotenuse is of length \(2x\).
Given that the side opposite the 60-degree angle is \(63 - \sqrt{?}\), we can set this equal to \(\sqrt{3}x\):
\[
\sqrt{3}x = 63 - \sqrt{?}
\]
To find \(x\), we can express \(x\) in terms of the given length of the side:
\[
x = \frac{63 - \sqrt{?}}{\sqrt{3}}
\]
Next, we calculate the hypotenuse:
\[
\text{hypotenuse} = 2x = 2 \cdot \frac{63 - \sqrt{?}}{\sqrt{3}} = \frac{2(63 - \sqrt{?})}{\sqrt{3}}
\]
Now we can simplify:
\[
\text{hypotenuse} = \frac{126 - 2\sqrt{?}}{\sqrt{3}}
\]
Though the exact numerical value of the hypotenuse relies on \( \sqrt{?}\), typically the final result matches one of the response options provided.
Still, without a specific numerical value for \( \sqrt{?} \) or further context, we cannot select the correct response. Please provide a specific value for the square root in question to find the hypotenuse definitively.
Here's what each side represents:
- The side opposite the 30-degree angle (shortest side) is of length \(x\).
- The side opposite the 60-degree angle is of length \(\sqrt{3}x\).
- The hypotenuse is of length \(2x\).
Given that the side opposite the 60-degree angle is \(63 - \sqrt{?}\), we can set this equal to \(\sqrt{3}x\):
\[
\sqrt{3}x = 63 - \sqrt{?}
\]
To find \(x\), we can express \(x\) in terms of the given length of the side:
\[
x = \frac{63 - \sqrt{?}}{\sqrt{3}}
\]
Next, we calculate the hypotenuse:
\[
\text{hypotenuse} = 2x = 2 \cdot \frac{63 - \sqrt{?}}{\sqrt{3}} = \frac{2(63 - \sqrt{?})}{\sqrt{3}}
\]
Now we can simplify:
\[
\text{hypotenuse} = \frac{126 - 2\sqrt{?}}{\sqrt{3}}
\]
Though the exact numerical value of the hypotenuse relies on \( \sqrt{?}\), typically the final result matches one of the response options provided.
Still, without a specific numerical value for \( \sqrt{?} \) or further context, we cannot select the correct response. Please provide a specific value for the square root in question to find the hypotenuse definitively.