A 10-foot-tall ladder rests against an 8-foot wall. The wall and ground form a 90-degree angle.

How many feet away from the wall is the bottom of the ladder?

1 answer

To solve this problem, we can visualize it as a right triangle where:

  • The height of the wall forms one leg of the triangle (8 feet).
  • The distance from the base of the wall to the bottom of the ladder forms the other leg (let's call this distance \( x \)).
  • The ladder itself forms the hypotenuse of the triangle (10 feet).

We can use the Pythagorean theorem, which states that in a right triangle:

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

where \( a \) and \( b \) are the legs of the triangle and \( c \) is the hypotenuse.

In our situation:

  • \( a = 8 \) feet (height of the wall),
  • \( b = x \) feet (distance from the wall),
  • \( c = 10 \) feet (length of the ladder).

We can substitute the values into the Pythagorean theorem:

\[ 8^2 + x^2 = 10^2 \]

Calculating the squares:

\[ 64 + x^2 = 100 \]

Next, we want to isolate \( x^2 \):

\[ x^2 = 100 - 64 \] \[ x^2 = 36 \]

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

\[ x = \sqrt{36} \] \[ x = 6 \]

Thus, the bottom of the ladder is 6 feet away from the wall.