Question

A right triangle has a hypotenuse of length 16 and an angle of 45°, with a side opposite this angle of length 82–√ . A second right triangle also has an angle of 45° and a side opposite this angle with a length of 42–√ . Determine the length of the hypotenuse in the second triangle.(1 point)
Responses

The hypotenuse of the second triangle has length 82–√.
The hypotenuse of the second triangle has length 8 Start Root 2 End Root .

The hypotenuse of the second triangle has length 42–√.
The hypotenuse of the second triangle has length 4 Start Root 2 End Root .

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

The hypotenuse of the second triangle has length 4

Answer 1

In a 45°-45°-90° triangle, the relationship between the lengths of the sides is characterized by the ratio:

- Each leg has length \( x \),
- The hypotenuse has length \( x\sqrt{2} \).

For the first triangle, we are given that:
- The length of the leg opposite the 45° angle is \( 82 - \sqrt{} \).

To find the hypotenuse of this triangle:
\[
\text{Hypotenuse} = (82 - \sqrt{})\sqrt{2}
\]

However, the focused value for the second triangle is:
- The length of the leg opposite the 45° angle is \( 42 - \sqrt{} \).

Now calculating the hypotenuse for the second triangle:
\[
\text{Hypotenuse} = (42 - \sqrt{})\sqrt{2}
\]

None of the options directly represent that value. However, considering the relationships between lengths, we note that for any side length \( a \), the length of the hypotenuse in a 45°-45°-90° triangle is always \( a\sqrt{2} \).

We will identify all possible responses:

1. **The hypotenuse of the second triangle has length \( \sqrt{82} \).**
2. **The hypotenuse of the second triangle has length \( 8\sqrt{2} \).**
3. **The hypotenuse of the second triangle has length \( \sqrt{42} \).**
4. **The hypotenuse of the second triangle has length \( 4\sqrt{2} \).**
5. **The hypotenuse of the second triangle has length \( 8. \)**
6. **The hypotenuse of the second triangle has length \( 4. \)**

Given that the hypotenuse for the second triangle can be expressed as \( (42 - \sqrt{})\sqrt{2} \), we can equate possible hypotenuse calculations. After comparing values (keeping \( \sqrt{2} \) in mind), the correct interpretation converts directly showing \( 4\sqrt{2} \) is valid based on the value substitutions appropriated.

Thus, confirming the hypotenuse of the second triangle is indeed:

The **hypotenuse of the second triangle has length \( 4\sqrt{2} \)**.