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To solve this problem, we need to use properties of 45-degree right triangles, where the sides follow a specific ratio due to the angles involved.

### Solution Steps:

1. **Identify the Given Triangle Properties**:
   - In a 45-45-90 triangle, the sides are in the ratio \(1 : 1 : \sqrt{2}\).
   - If one of the legs is \( x \), the hypotenuse will be \( x \sqrt{2} \).

2. **Analyze the First Triangle**:
   - The first triangle has a hypotenuse of \(10\sqrt{2}\) and a 45-degree angle.
   - Since this is a 45-45-90 triangle, the legs are equal.
   - Let each leg be \( x \). Then, by the 45-45-90 triangle property, we have:
     \[
     x\sqrt{2} = 10\sqrt{2}
     \]
   - Solving for \( x \), we divide both sides by \( \sqrt{2} \):
     \[
     x = 10
     \]
   - So, each leg of this triangle is 10.

3. **Set Up the Second Triangle**:
   - The second triangle is also a 45-45-90 triangle with a leg (opposite the 45-degree angle) of length 14.
   - Since the two legs are equal, the other leg is also 14.

4. **Find the Hypotenuse of the Second Triangle**:
   - Using the 45-45-90 triangle ratio, the hypotenuse \( h \) is:
     \[
     h = 14 \sqrt{2}
     \]

### Conclusion:
The hypotenuse of the second triangle has length \(14\sqrt{2}\).

### Answer:
The correct answer is:
- **The hypotenuse of the second triangle has length \(14\sqrt{2}\)**.

Would you like a more detailed explanation or have further questions?

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#### Related Questions:
1. How do we calculate the hypotenuse in any 45-45-90 triangle?
2. Why do 45-45-90 triangles have sides in a \(1:1:\sqrt{2}\) ratio?
3. What would the legs' lengths be if the hypotenuse of a 45-45-90 triangle were 20?
4. Can we apply the Pythagorean theorem directly to verify the solution here?
5. How would the solution differ if the angle was not 45 degrees?

#### Tip:
In any 45-45-90 triangle, knowing one leg length immediately gives the other leg (same length) and the hypotenuse (that length times \( \sqrt{2} \)). This property is unique to 45-45-90 triangles.

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.

Solve a problem involving two right triangles with a 45-degree angle. Given a leg length of 14 in the second triangle, learn how to calculate the hypotenuse using the properties of 45-45-90 triangles. Ideal for students in Grades 9-10.

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