https://file.aiquickdraw.com/mathbot/file/1730399171194iq6jxg0e.jpg

To solve this problem, we need to analyze the properties of a 45-45-90 right triangle.

### Step-by-Step Solution

In a 45-45-90 triangle:
- Both legs are equal in length.
- The hypotenuse is \( \text{leg} \times \sqrt{2} \).

#### Triangle 1
- The hypotenuse of the first triangle is given as \( 10\sqrt{2} \).
- Using the 45-45-90 triangle properties, each leg \( = \frac{\text{hypotenuse}}{\sqrt{2}} \).
  
So, for the first triangle:
\[
\text{leg} = \frac{10\sqrt{2}}{\sqrt{2}} = 10
\]

This matches the information given in the problem (one leg has a length of 10).

#### Triangle 2
- The second triangle has one leg opposite the 45° angle with a length of 14.
- To find the hypotenuse, we again use the 45-45-90 triangle rule: hypotenuse = leg \( \times \sqrt{2} \).

So, for the second triangle:
\[
\text{hypotenuse} = 14 \times \sqrt{2} = 14\sqrt{2}
\]

### Conclusion
The correct answer is:
\[
\text{The hypotenuse of the second triangle has length } 14\sqrt{2}.
\]

This corresponds to the first answer choice:
- **The hypotenuse of the second triangle has length \( 14\sqrt{2} \).**

Would you like further details, or do you have any questions?

### Related Questions
1. How do we apply the 45-45-90 rule to different triangles?
2. Why does a 45-45-90 triangle have the hypotenuse as \( \text{leg} \times \sqrt{2} \)?
3. How would the solution change if the angle were different, say 30°?
4. What if the hypotenuse of the first triangle were different?
5. Can we generalize this approach to find missing lengths in any right triangle?

### Tip
In a 45-45-90 triangle, remember that both legs are always equal, and the hypotenuse is \( \sqrt{2} \) times any leg.

A right triangle has a hypotenuse of length 10√2 and an angle of 45 degrees, with a side opposite this angle with a length of 10. A second right triangle also has an angle of 45 degrees, with a side opposite this angle with a length of 14. Determine the length of the hypotenuse in the second triangle.

This geometry problem involves finding the hypotenuse of a 45-45-90 triangle given one leg's length. Learn the step-by-step approach using special triangle properties, suitable for Grades 9-10.

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