How high is the wall?

Question

a ladder is two feet longer than the height of the wall. when the top of the ladder is placed against the top of the wall, the distance from the base of the ladder to the wall is equal to the height of the wall. how high is the wall?

Answer 1

Let h be the height of the wall in feet and x the length of the ladder, also in feet.

First we have: x = h + 2.

When the top of the ladder is placed against the top of the wall a right triangle forms, with x the hypotenuse and h the vertical leg.

Now the distance from the base of the ladder to the wall is equal to the height of the wall. This means the other leg , the horizontal one, measures the same as h.
Pythagoras' Theorem tells us that x² = h² + h² = 2h².

But, since x = h + 2 we have:

(h+2)² = 2h²

or

(h² + 4h + 4) = 2h²

or

h² - 4h - 4 = 0

This quadratic equation has solutions: 

2 - 2√2 which is negative and

2 + 2√2 which is positive and the right answer: approximately 4.83

 

Hope this helps!