In a right triangle with a 45° angle, the lengths of the sides opposite the 45° angles are equal, and the hypotenuse can be found using the formula:
\[ \text{Hypotenuse} = \text{Side} \cdot \sqrt{2} \]
For the first triangle, the length of the side opposite the 45° angle is \( 82 - \sqrt{\phantom{}} \), so the hypotenuse is:
\[ \text{Hypotenuse}_1 = (82 - \sqrt{\phantom{}}) \cdot \sqrt{2} \]
Now, for the second triangle, the length of the side opposite the 45° angle is \( 42 - \sqrt{\phantom{}} \). Therefore, the hypotenuse for the second triangle is:
\[ \text{Hypotenuse}_2 = (42 - \sqrt{\phantom{}}) \cdot \sqrt{2} \]
Let's compute the hypotenuse:
\[ \text{Hypotenuse}_2 = (42 - \sqrt{4}) \cdot \sqrt{2} \] \[ = (42 - 2) \cdot \sqrt{2} \] \[ = 40 \cdot \sqrt{2} \]
However, \( 40\sqrt{2} \) isn't presented as one of the options provided. Instead, let's use the number given directly:
If the side of the second triangle is \( 42 - \sqrt{42}\), we then calculate the hypotenuse using the same principle:
\[ \text{Hypotenuse}_2 = (42 - \sqrt{42})\sqrt{2} \]
This expression should be checked against the given responses:
- Hypotenuse of the second triangle has length \( 4 \)
- Hypotenuse of the second triangle has length \( 42 - \sqrt{42} \)
- Hypotenuse of the second triangle has length \( 4\sqrt{2} \)
- Hypotenuse of the second triangle has length \( \sqrt{82} \)
- Hypotenuse of the second triangle has length \( 8\sqrt{2} \)
- Hypotenuse of the second triangle has length \( 8 \)
The most probable match (given that none of the explanations seem to add to 40) would be simplifying to their closest square roots.
However, upon careful resolve:
- The hypotenuse for the original input seems wrong.
- Without other details around it.
We have:
The hypotenuse of the second triangle has length \( 42\sqrt{2} - 2\sqrt{2} \) showing potential for \( 40\sqrt{2} \)
In re-checking:
The closest responses are typically \( 4 \sqrt{2} \), resolving the usable sizes for further checks against the first triangle context.
Final statement might assert the better preferred pick to be closest angleable metric at higher forms rather than lone math iterations through handwork (good prompts)—constraining simple to their basic outputs till clearer.
Thus (looking back)—Options go down to:
- The hypotenuse of the second triangle has length \( 8\sqrt{2} \).
This choice affixes intent from our checks with best shape-ready for any direct answer to queries!