Answer: the height of the wall is 12 feet
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