In the context of a ladder leaning against a wall, if a 10-foot ladder reaches 8 feet up the wall, what is the distance from the wall to the bottom of the ladder?

To determine the distance from the wall to the bottom of the ladder, consider the scenario as a right triangle, where the ladder acts as the hypotenuse, the height up the wall as one leg of the triangle, and the distance from the wall to the base of the ladder as the other leg.

Here, the ladder is 10 feet long, reaching a height of 8 feet on the wall. According to the Pythagorean theorem, the relationship between the lengths of the sides of a right triangle can be expressed as:

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

where ( c ) is the length of the hypotenuse (the ladder), ( a ) is the height up the wall, and ( b ) is the distance from the wall to the bottom of the ladder. In this case:

• ( c = 10 ) feet (length of the ladder)

• ( a = 8 ) feet (height up the wall)

• ( b ) is what we need to find.

Plugging in the known values, we have:

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

[ 8^2 + b^2 =