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

Answer 1

In a right triangle with a 45° angle, the two legs are of equal length. We denote the length of each leg as \( x \). The length of the hypotenuse \( h \) of such a triangle can be determined using the formula:

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

First Triangle:

The first triangle is given:

• Hypotenuse \( h_1 = 16 \)
• Opposite side of the 45° angle \( x_1 = 8\sqrt{2} \)

To verify if the values are consistent, we note that in this triangle both legs are equal since it’s a 45°-45°-90° triangle. Therefore, we have:

\[ x_1 = x_2 = 8\sqrt{2} \]

Calculating the hypotenuse:

\[ h_1 = x_1 \sqrt{2} = 8\sqrt{2} \cdot \sqrt{2} = 8 \cdot 2 = 16 \]

This checks out with the given hypotenuse of 16.

Second Triangle:

For the second triangle, we are given:

• Opposite side of the 45° angle \( x_2 = 4\sqrt{2} \)

Now, we can find the hypotenuse \( h_2 \) of this second triangle using the same formula:

\[ h_2 = x_2 \sqrt{2} = 4\sqrt{2} \cdot \sqrt{2} = 4 \cdot 2 = 8 \]

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

\[ \boxed{8} \]