Question

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

The hypotenuse of the second triangle has length 4 square root of 2.

The hypotenuse of the second triangle has length 4.

The hypotenuse of the second triangle has length 8 square root of 2.

The hypotenuse of the second triangle has length 8.

Answer 1

To solve the problem, we need to analyze the properties of a right triangle with an angle of 45 degrees.

In a 45-45-90 triangle, the sides are in the ratio \(1:1:\sqrt{2}\). This means that if the lengths of the legs are both \(x\), then the hypotenuse \(h\) is given by:

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

Given the second triangle, we have a side opposite the 45-degree angle with a length of \(4\sqrt{2}\). In this case, since this side is one of the legs of the triangle, we can set it as \(x\). Therefore:

\[
x = 4\sqrt{2}
\]

To find the length of the hypotenuse \(h\), we can apply the formula mentioned above:

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

Calculating \(h\),

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

Thus, the hypotenuse of the second triangle has a length of \(8\).

The correct answer is:

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