Question

What is the length of the missing leg? One leg is 6.5 ft and the slope is 7.5 ft. If necessary, round to the nearest tenth.

Answer 1

To find the length of the missing leg of a right triangle when you know one leg and the length of the hypotenuse, you can use the Pythagorean theorem. The theorem states:

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

Where:
- \( a \) is one leg of the triangle,
- \( b \) is the other leg of the triangle (the one we want to find),
- \( c \) is the hypotenuse (the slope in this case).

Let’s assign the given values:
- One leg \( a = 6.5 \) ft
- Hypotenuse \( c = 7.5 \) ft

We need to find \( b \).

Rearranging the Pythagorean theorem to solve for \( b \):

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

Substituting in the known values:

\[
b^2 = (7.5)^2 - (6.5)^2
\]

Calculating \( (7.5)^2 \) and \( (6.5)^2 \):

\[
(7.5)^2 = 56.25
\]
\[
(6.5)^2 = 42.25
\]

Now, substituting these values back into the equation:

\[
b^2 = 56.25 - 42.25
\]
\[
b^2 = 14
\]

Now take the square root to find \( b \):

\[
b = \sqrt{14} \approx 3.74
\]

Rounding to the nearest tenth, the length of the missing leg is:

\[
\boxed{3.7} \text{ ft}
\]