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.