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.

A. The hypotenuse of the second triangle has length 8 square root 2.
B. The hypotenuse of the second triangle has length 8.
C. The hypotenuse of the second triangle has length 4 square root 2.
D. The hypotenuse of the second triangle has length 4.

Answer 1

Thank You!

Answer 2

In a 45-degree right triangle, the properties of the triangle are such that the lengths of the two legs are equal, and the relationship between the length of a leg \( a \) and the hypotenuse \( c \) can be given by the formula:

\[
c = a \sqrt{2}
\]

Let's first analyze the second triangle which has a side opposite the 45-degree angle of length \( 4\sqrt{2} \). Since this is a 45-degree right triangle, the lengths of both legs are equal, meaning the other leg is also \( 4\sqrt{2} \).

Now, we can find the length of the hypotenuse \( c \):

\[
c = a \sqrt{2}
\]
Substituting \( a = 4\sqrt{2} \):

\[
c = (4\sqrt{2}) \sqrt{2}
\]
\[
c = 4 \cdot 2
\]
\[
c = 8
\]

Now we can conclude that the hypotenuse of the second triangle has a length of \( 8 \).

Thus, the correct answer is:

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