Question
If the side opposite a 60-degree angle in a 30-60-90 right triangle is 73–√ cm
long, how long is the hypotenuse?(1 point)
Responses
28 cm
28 cm
73–√ cm
7 Start Root 3 End Root cm
7 cm
7 cm
14 cm
long, how long is the hypotenuse?(1 point)
Responses
28 cm
28 cm
73–√ cm
7 Start Root 3 End Root cm
7 cm
7 cm
14 cm
Answers
GPT-4o mini
In a 30-60-90 right triangle, the ratios of the lengths of the sides are as follows:
- The side opposite the 30-degree angle (shortest side) is \( x \).
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
- The hypotenuse is \( 2x \).
Given that the side opposite the 60-degree angle is \( 73 - \sqrt{cm} \), we can set up the following equation:
\[
x\sqrt{3} = 73 - \sqrt{cm}
\]
To find \( x \):
\[
x = \frac{73 - \sqrt{cm}}{\sqrt{3}}
\]
Now, using the ratio for the hypotenuse:
\[
\text{Hypotenuse} = 2x = 2 \left(\frac{73 - \sqrt{cm}}{\sqrt{3}}\right) = \frac{2(73 - \sqrt{cm})}{\sqrt{3}}
\]
Now it's essential to observe what is being asked in the responses. Since the responses are numerical values, we need to determine what value we could reasonably expect as an answer based on our derived formulas.
Let's evaluate the relationship. Given that \( x\sqrt{3} = 73 - \sqrt{cm} \) implies that \( (73 - \sqrt{cm}) \) must be a realizable side length for a triangle. If we assume that the side opposite \( 30^\circ \) is the main value we can work with, we can test for some hypothetical lengths that align with \( 73 - \sqrt{cm} \).
To return to the question, the responses are numerical values. It seems we need to look closely at the actual numeric options instead of continuing with algebraic representation.
From the given numbers:
- Since the hypotenuse is twice the length of the side \( x \), if we make \( x\sqrt{3} = 73 - \sqrt{cm} \), we will need reasonable \( x \) values that match the defined answers.
Finally, among the responses—28 cm stands out as one possible valid length for a hypotenuse.
Thus, the length of the hypotenuse, given \( x\sqrt{3} = 73 - \sqrt{cm} \), can potentially produce:
**The length of the hypotenuse is 28 cm.**
- The side opposite the 30-degree angle (shortest side) is \( x \).
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
- The hypotenuse is \( 2x \).
Given that the side opposite the 60-degree angle is \( 73 - \sqrt{cm} \), we can set up the following equation:
\[
x\sqrt{3} = 73 - \sqrt{cm}
\]
To find \( x \):
\[
x = \frac{73 - \sqrt{cm}}{\sqrt{3}}
\]
Now, using the ratio for the hypotenuse:
\[
\text{Hypotenuse} = 2x = 2 \left(\frac{73 - \sqrt{cm}}{\sqrt{3}}\right) = \frac{2(73 - \sqrt{cm})}{\sqrt{3}}
\]
Now it's essential to observe what is being asked in the responses. Since the responses are numerical values, we need to determine what value we could reasonably expect as an answer based on our derived formulas.
Let's evaluate the relationship. Given that \( x\sqrt{3} = 73 - \sqrt{cm} \) implies that \( (73 - \sqrt{cm}) \) must be a realizable side length for a triangle. If we assume that the side opposite \( 30^\circ \) is the main value we can work with, we can test for some hypothetical lengths that align with \( 73 - \sqrt{cm} \).
To return to the question, the responses are numerical values. It seems we need to look closely at the actual numeric options instead of continuing with algebraic representation.
From the given numbers:
- Since the hypotenuse is twice the length of the side \( x \), if we make \( x\sqrt{3} = 73 - \sqrt{cm} \), we will need reasonable \( x \) values that match the defined answers.
Finally, among the responses—28 cm stands out as one possible valid length for a hypotenuse.
Thus, the length of the hypotenuse, given \( x\sqrt{3} = 73 - \sqrt{cm} \), can potentially produce:
**The length of the hypotenuse is 28 cm.**