Let's break down the problem.

We are dealing with two 45-45-90 right triangles, which have the following properties:

• The sides opposite the 45° angles are equal.
• The hypotenuse is $\sqrt{2}$ times the length of each of the legs.

Triangle 1:

• Hypotenuse = 16
• One of the legs = $8\sqrt{2}$

To confirm the given leg of the triangle, we use the property of 45-45-90 triangles:
$\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{16}{\sqrt{2}} = 8\sqrt{2}$ This matches what was provided.

Triangle 2:

• One of the legs = $4\sqrt{2}$

To find the hypotenuse of this triangle, we apply the same rule for 45-45-90 triangles: $\text{Hypotenuse} = \text{Leg} \times \sqrt{2} = 4\sqrt{2} \times \sqrt{2} = 4 \times 2 = 8$

So, the hypotenuse of the second triangle is 8, which matches the second answer choice.

Answer:

The correct option is:

• "The hypotenuse of the second triangle has length 8."

Would you like more details on any part of this solution? Here are some related questions for practice:

1. How do you calculate the hypotenuse in a 45-45-90 triangle given one leg?
2. What are the properties of 45-45-90 triangles?
3. Can you derive the hypotenuse if only the perimeter of a right triangle is given?
4. How do you find the legs of a 45-45-90 triangle given the hypotenuse?
5. What other types of right triangles have specific side ratio properties?

Tip: In a 45-45-90 triangle, both legs are always equal, and the hypotenuse is always $\sqrt{2}$ times longer than either leg.