https://file.aiquickdraw.com/mathbot/file/1730081610721lpmubuxj.jpg

To find the length of the missing hypotenuse in the larger triangle, we can observe that both triangles shown are similar right triangles.

### Steps:
1. **Identify the smaller triangle**:
   - The smaller triangle is a 45-45-90 triangle with legs of length 1 and hypotenuse of \( \sqrt{2} \).

2. **Use the properties of similar triangles**:
   - In a 45-45-90 triangle, the legs are equal, and the hypotenuse is \( \sqrt{2} \) times the length of each leg.
   - The larger triangle is also a 45-45-90 triangle since its legs are both equal (6).

3. **Set up the proportion**:
   - The ratio of the side lengths in any 45-45-90 triangle is \( 1 : 1 : \sqrt{2} \).
   - Since each leg of the larger triangle is 6, the hypotenuse of the larger triangle will be \( 6 \times \sqrt{2} \).

### Conclusion:
The length of the missing hypotenuse is:
\[
6\sqrt{2}
\]

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

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### Related Questions:
1. How would you find the hypotenuse of a right triangle if only one leg is given?
2. What is the relationship between the sides of a 30-60-90 triangle?
3. How can similarity of triangles be used to solve for unknown lengths?
4. Why is the ratio \(1 : 1 : \sqrt{2}\) specific to 45-45-90 triangles?
5. How would you calculate the hypotenuse if the legs had different lengths?

### Tip:
In any 45-45-90 triangle, if you know the length of one leg, you can immediately determine the hypotenuse by multiplying the leg length by \( \sqrt{2} \).

Give an expression for the length of the missing hypotenuse.

Learn how to find the length of the hypotenuse in a 45-45-90 triangle with given leg lengths. This problem explores concepts of similar triangles, special right triangles, and uses the formula Hypotenuse = leg * √2. Suitable for students in Grades 8-10.

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