Question

The bottom of a ladder must be placed 5 ft. from a wall. The ladder is 12 feet long.


Based on the diagram, what is the distance in feet from the ground that the ladder reaches the wall? Round to the nearest tenth.

Answer 1

To find the distance that the ladder reaches the wall, we can use the Pythagorean theorem. The setup can be represented as a right triangle, where:

- The length of the ladder is the hypotenuse (12 feet),
- The distance from the wall is one leg of the triangle (5 feet), and
- The height at which the ladder touches the wall is the other leg we need to find.

Let's denote the height at which the ladder touches the wall as \( h \). According to the Pythagorean theorem:

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

where:
- \( a \) is one leg (the distance from the wall, which is 5 feet),
- \( b \) is the other leg (the height \( h \) we want to find),
- \( c \) is the hypotenuse (the length of the ladder, which is 12 feet).

Plugging in the values, we get:

\[
5^2 + h^2 = 12^2
\]

Calculating the squares:

\[
25 + h^2 = 144
\]

Now, solving for \( h^2 \):

\[
h^2 = 144 - 25
\]
\[
h^2 = 119
\]

Now take the square root of both sides to find \( h \):

\[
h = \sqrt{119} \approx 10.9087
\]

Rounding to the nearest tenth, the height at which the ladder reaches the wall is approximately:

\[
\boxed{10.9} \text{ feet}
\]