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
14 cm
14 cm
28 cm
28 cm
73–√ cm
7 Start Root 3 End Root cm
7 cm
Responses
14 cm
14 cm
28 cm
28 cm
73–√ cm
7 Start Root 3 End Root cm
7 cm
Answers
GPT-4o mini
In a 30-60-90 triangle, the lengths of the sides are in the ratio \(1 : \sqrt{3} : 2\). Specifically, if the length of the side opposite the 30-degree angle (the shorter leg) is \(x\), then:
- The length of the hypotenuse is \(2x\).
- The length of the side opposite the 60-degree angle (the longer leg) is \(x\sqrt{3}\).
In this case, the side opposite the 60-degree angle is given as \(73 - \sqrt{cm}\).
Using the relationship for a 30-60-90 triangle, we have:
\[
x\sqrt{3} = 73 - \sqrt{cm}
\]
To find \(x\), we can rearrange:
\[
x = \frac{73 - \sqrt{cm}}{\sqrt{3}}
\]
Now, we need to find the hypotenuse:
\[
\text{Hypotenuse} = 2x = 2 \left(\frac{73 - \sqrt{cm}}{\sqrt{3}}\right) = \frac{2(73 - \sqrt{cm})}{\sqrt{3}}
\]
To find the value of the hypotenuse in terms of cm, since the side lengths you've mentioned seem to be numerical rather than variable, we can simplify further:
If we assume \(x\) produces a valid length when substituting, we focus on the simplest calculation. Given \(x\) relates to \(73 - \sqrt{cm}\), and the calculations often yield whole numbers.
From the possible answers provided, we can evaluate:
1. If \(x = 73 - \sqrt{cm}\) equals the 30-degree side, the hypotenuse will land accurately based on normalized estimates or if \(x\) approximates a common rational denominator scaling.
2. Suppose we want ordinary outcomes relating directly to the actual calculated estimates leading back to a basic estimate.
Let's verify by setting safe constant estimations, keep the portion integrity.
Based on our understanding above, direct calculation leading hypotenuse guessing gives:
- The best-fit answer deriving approximates to **28 cm** being valid to explore based on calculative scenarios provided!
Thus, the hypotenuse is **28 cm**.
- The length of the hypotenuse is \(2x\).
- The length of the side opposite the 60-degree angle (the longer leg) is \(x\sqrt{3}\).
In this case, the side opposite the 60-degree angle is given as \(73 - \sqrt{cm}\).
Using the relationship for a 30-60-90 triangle, we have:
\[
x\sqrt{3} = 73 - \sqrt{cm}
\]
To find \(x\), we can rearrange:
\[
x = \frac{73 - \sqrt{cm}}{\sqrt{3}}
\]
Now, we need to find the hypotenuse:
\[
\text{Hypotenuse} = 2x = 2 \left(\frac{73 - \sqrt{cm}}{\sqrt{3}}\right) = \frac{2(73 - \sqrt{cm})}{\sqrt{3}}
\]
To find the value of the hypotenuse in terms of cm, since the side lengths you've mentioned seem to be numerical rather than variable, we can simplify further:
If we assume \(x\) produces a valid length when substituting, we focus on the simplest calculation. Given \(x\) relates to \(73 - \sqrt{cm}\), and the calculations often yield whole numbers.
From the possible answers provided, we can evaluate:
1. If \(x = 73 - \sqrt{cm}\) equals the 30-degree side, the hypotenuse will land accurately based on normalized estimates or if \(x\) approximates a common rational denominator scaling.
2. Suppose we want ordinary outcomes relating directly to the actual calculated estimates leading back to a basic estimate.
Let's verify by setting safe constant estimations, keep the portion integrity.
Based on our understanding above, direct calculation leading hypotenuse guessing gives:
- The best-fit answer deriving approximates to **28 cm** being valid to explore based on calculative scenarios provided!
Thus, the hypotenuse is **28 cm**.