In a 45°-45°-90° triangle, the relationship between the lengths of the sides and the hypotenuse is as follows:
- If the lengths of the legs (the sides opposite the 45° angles) are \( x \), then the length of the hypotenuse \( h \) is given by: \[ h = x\sqrt{2} \]
For the first triangle:
- We have a side opposite the 45° angle of length \( 82 - \sqrt{8} \).
- The hypotenuse \( h_1 \) can be calculated as: \[ h_1 = (82 - \sqrt{8})\sqrt{2} \]
Now let's analyze the second triangle:
- The length of the side opposite the 45° angle is \( 42 - \sqrt{4} \).
- The length \( \sqrt{4} \) simplifies to \( 2 \), so the length of this side is \( 42 - 2 = 40 \).
- Therefore, the hypotenuse \( h_2 \) can be calculated as: \[ h_2 = (42 - \sqrt{4})\sqrt{2} = (40)\sqrt{2} \]
To determine the numerical value of \( h_2 \):
- Calculating \( h_2 \): \[ h_2 = 40\sqrt{2} \]
So, the lengths of the hypotenuses for the second triangle are as follows:
- The hypotenuse of the second triangle has length \( 40\sqrt{2} \).
Unfortunately, this specific value \( 40\sqrt{2} \) is not included in the original answer choices listed. Please verify the options provided to confirm if any correspond directly to \( 40\sqrt{2} \). However, amongst the provided choices, it appears they do not match. The hypotenuse length is \[ 40\sqrt{2} \].
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