A 25-foot ladder leans against a wall so that it is 20 feet high at the top. The ladder is moved so that the base of the ladder travels toward the wall twice the distance that the top of the ladder moves up. How much higher is the top of the ladder now?
Here is a before and after drawing of the scenario:
The question is what is y?
First we can solve for x.
Use Pythagorean Theorem.
x2 + 202 = 252
x = 15
Now use this info about x to solve for y with Pythagorean Theorem..
(15 - 2y)2 + (20 + y)2 = 252
Multiply this out.
(15 - 2y)(15 - 2y) + (20 + y)(20 + y) = 625
225 - 60y + 4y2 + 400 + 40y + y2 = 625
Combine like terms and write in descending order.
5y2 - 20y + 625 = 625
5y2 - 20y = 0
Divide everything by 5.
y2 - 4y = 0
(y)(y-4) = 0
Set each factor equal to zero and solve for y.
Either y = 0 or y = 4
It doesn't make sense for y to be 0, if y was 0, it would be the exact same triangle we started with!
y = 4 feet