One end of a 13-foot-long ladder is resting on the top of a vertical wall. The distance from the foot of the ladder to the base of the wall is 7 feet less than the height of the wall. How high is the wall? Suggestion: Recall the Pythagorean theorem, which says for legs a and b and hypotenuse c of a right triangle that?

Answer: the height of the wall is 12 feet

Step-by-step explanation:

The ladder forms a right angle triangle with the vertical wall and the ground. The length of the ladder represents the hypotenuse, c of the right angle triangle. The height from the top of the ladder to the base of the vertical wall represents the leg, a of the right angle triangle.

The distance from the bottom of the ladder to the base of the vertical wall represents the leg, b side of the right angle triangle.

The distance from the foot of the ladder to the base of the wall is 7 feet less than the height of the wall. This means that

a = b + 7

To determine the height of the wall, a we would apply Pythagoras theorem which is expressed as

Hypotenuse² = leg a² + leg b²

13² = (b + 7) ² + b²

169 = b² + 14b + 49 + b²

2b² + 14b + 49 - 169 = 0

2b² + 14b - 120 = 0

Dividing through by 2, it becomes

b² + 7b - 60 = 0

b² + 12b - 5b - 60 = 0

b (b + 12) - 5 (b + 12)

b - 5 = 0 or b + 12 = 0

b = 5 or b = - 12

The distance cannot be negative so b = 5

a = b + 7 = 5 + 7

a = 12