To determine the length of the missing leg in a right triangle given the lengths of the other two legs (91 miles and 84 miles), we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
Where:
- \(c\) is the length of the hypotenuse
- \(a\) and \(b\) are the lengths of the other two sides.
In this case, we need to find the length of either leg \(b\) when one leg \(a\) is 91 miles and the other leg is 84 miles:
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Assign numbers:
- Let \(a = 91\) miles
- Let \(b = 84\) miles
- Then, square both sides: \[ 91^2 + 84^2 = c^2 \]
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Calculate the squares:
- \(91^2 = 8281\)
- \(84^2 = 7056\)
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Add the squares: \[ 8281 + 7056 = c^2 \] \[ 15337 = c^2 \]
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Now, take the square root to find \(c\): \[ c = \sqrt{15337} \approx 123.9 \text{ miles} \]
If your question refers to finding the other leg if one leg is missing, please clarify the scenario. However, if the legs' lengths are interchanged, but we still calculate for both leg lengths, both legs can be considered interchangeable with respect to the hypotenuse.
From the above work, the hypotenuse \(c\) is approximately \(123.9\) miles.
If, instead, you were looking for the missing leg assuming one of the legs was missing, please clarify the scenario of what specifically is needed.