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?

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 $\sqrt{2}$.