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 7 cm 7 cm 73–√ cm 7 Start Root 3 End Root cm 28 cm 28 cm 14 cm
Answers
GPT-4o mini
In a 30-60-90 right triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where:
- The side opposite the 30-degree angle 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 this equal to \(x\sqrt{3}\):
\[
x\sqrt{3} = 73 - \sqrt{cm}
\]
To find \(x\), we can solve for \(x\):
\[
x = \frac{73 - \sqrt{cm}}{\sqrt{3}}
\]
Now, the hypotenuse, which is \(2x\), can be calculated:
\[
\text{Hypotenuse} = 2x = 2\left(\frac{73 - \sqrt{cm}}{\sqrt{3}}\right) = \frac{2(73 - \sqrt{cm})}{\sqrt{3}}
\]
Since we need to find the hypotenuse numerically or directly from the options provided, we can evaluate it with an approximate assumption or simplification if needed.
However, without further specifics or alternative evaluations, I notice that the options include numerical values not closely matching this derived relationship. Nevertheless, from option observation, we can test numeric values such as \(28\) and simplify as necessary.
Let’s check the hypotenuse based on logical comparisons:
1. If \(x = 7\), we calculate:
- \(x\sqrt{3} = 7\sqrt{3}\) unleverages immediate comparisons.
- Thus check directly with values and ratios if numerically relevant.
The ideal relation hints it corresponds more closely to commands like translate or numeric ratios, ultimately it's understood that:
Given the set circumstances and problem stipulation of triangle geometries derivatively, the derived hypotenuse has typically manageable characteristics. Therefore, with rational values available, we are aligning broadly with the relational hypotenuse forming \(28\) would likely appear most suitably close.
Thus, the hypotenuse, given outcomes, should thus conclude as:
\[
\text{Hypotenuse length} = 28 \text{ cm}
\]
Conclusively, the resultant length of the hypotenuse is **28 cm**.
- The side opposite the 30-degree angle 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 this equal to \(x\sqrt{3}\):
\[
x\sqrt{3} = 73 - \sqrt{cm}
\]
To find \(x\), we can solve for \(x\):
\[
x = \frac{73 - \sqrt{cm}}{\sqrt{3}}
\]
Now, the hypotenuse, which is \(2x\), can be calculated:
\[
\text{Hypotenuse} = 2x = 2\left(\frac{73 - \sqrt{cm}}{\sqrt{3}}\right) = \frac{2(73 - \sqrt{cm})}{\sqrt{3}}
\]
Since we need to find the hypotenuse numerically or directly from the options provided, we can evaluate it with an approximate assumption or simplification if needed.
However, without further specifics or alternative evaluations, I notice that the options include numerical values not closely matching this derived relationship. Nevertheless, from option observation, we can test numeric values such as \(28\) and simplify as necessary.
Let’s check the hypotenuse based on logical comparisons:
1. If \(x = 7\), we calculate:
- \(x\sqrt{3} = 7\sqrt{3}\) unleverages immediate comparisons.
- Thus check directly with values and ratios if numerically relevant.
The ideal relation hints it corresponds more closely to commands like translate or numeric ratios, ultimately it's understood that:
Given the set circumstances and problem stipulation of triangle geometries derivatively, the derived hypotenuse has typically manageable characteristics. Therefore, with rational values available, we are aligning broadly with the relational hypotenuse forming \(28\) would likely appear most suitably close.
Thus, the hypotenuse, given outcomes, should thus conclude as:
\[
\text{Hypotenuse length} = 28 \text{ cm}
\]
Conclusively, the resultant length of the hypotenuse is **28 cm**.