Question

A right triangle has a hypotenuse of length 16 and an angle of 45 degrees, with a side opposite this angle of length 8 square root 2. A second right triangle also has an angle of 45 degrees and a side opposite this angle with a length of 4 square root 2. Determine the length of the hypotenuse in the second triangle.

Answer 1

To solve for the hypotenuse of the second triangle, we first recognize that both triangles are isosceles right triangles (45-45-90 triangles). In such triangles, the relationships between the lengths of the sides and the hypotenuse are consistent: if the legs (the two sides opposite the 45° angles) have length \( x \), the hypotenuse \( h \) is given by

\[
h = x \sqrt{2}.
\]

Now, we analyze the second triangle:

1. **Identifying the leg length**: The length of the side opposite the 45° angle in the second triangle is \( 4 \sqrt{2} \).

2. **Calculating the hypotenuse**: Since the length of each leg in a 45-45-90 triangle is equal, we can identify the leg length \( x \) as \( 4 \sqrt{2} \).

To find the hypotenuse \( h \), we apply the formula:

\[
h = x \sqrt{2}.
\]
Substituting \( x = 4 \sqrt{2} \):

\[
h = (4 \sqrt{2}) \sqrt{2}.
\]

Using the property of square roots \( \sqrt{2} \cdot \sqrt{2} = 2 \), we simplify this as:

\[
h = 4 \cdot 2 = 8.
\]

Thus, the length of the hypotenuse in the second triangle is

\[
\boxed{8}.
\]