Question

What is the length of the unknown leg in the right triangle?

A right triangle has a side with length 9 miles, hypotenuse with length StartRoot 113 EndRoot miles, and side labeled a.
StartRoot 22 EndRoot mi
StartRoot 32 EndRoot mi
StartRoot 104 EndRoot mi
StartRoot 194 EndRoot mi

Answer 1

To solve for the length of the unknown leg \( a \) in the right triangle, we can use the Pythagorean theorem:

\[
a^2 + b^2 = c^2
\]

where:
- \( a \) is the length of the unknown leg,
- \( b = 9 \) miles (the length of the other leg),
- \( c = \sqrt{113} \) miles (the length of the hypotenuse).

Substituting \( b \) and \( c \) into the Pythagorean theorem gives:

\[
a^2 + 9^2 = (\sqrt{113})^2
\]

Calculating \( 9^2 \) and \( (\sqrt{113})^2 \):

\[
a^2 + 81 = 113
\]

Now, we can isolate \( a^2 \):

\[
a^2 = 113 - 81
\]

\[
a^2 = 32
\]

Now, we take the square root of both sides to find \( a \):

\[
a = \sqrt{32}
\]

We can simplify \( \sqrt{32} \):

\[
\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}
\]

Thus, the length of the unknown leg \( a \) is:

\[
\sqrt{32} \text{ miles or } 4\sqrt{2} \text{ miles}
\]

Based on the options you provided, the closest match is:

\[
\sqrt{32} \, \text{miles}
\]

So the answer is:

\[
\sqrt{32} \text{ miles}
\]