A ladder leans against a house with its base 15 feet from the house. When the ladder is pulled 9 feet farther away from the house, the upper end of the ladder slides down 13 feet. How long is the ladder?
See the diagram below.
the first triangle represent s the ladder's initial position. we are not sure what height on the wall is the top of the ladder, so we call it x. the base is 15 away from the house, so the base of the right triangle is 15. By Pythagoras, the hypotenuse (which is the length of the ladder) is $\displaystyle \sqrt {x^2 + 15^2}$
the second triangle represents the new position. the foot of the ladder is pull an additional 9 feet from the house, so the new base is 15 + 9 = 24, the height of the ladder on the wall decreases by 13, so the new height is x - 13. again, by Pythagoras, the length of the ladder is $\displaystyle \sqrt {(x - 3)^2 + 24^2}$
Now the length of the ladder does not change, so we just equate the two formulas obtained.
$\displaystyle \sqrt {x^2 + 15^2} = \sqrt {(x - 3)^2 + 24^2}$
$\displaystyle \Rightarrow x^2 + 15^2 = (x - 3)^2 + 24^2$
$\displaystyle \Rightarrow x^2 + 225 = x^2 - 6x + 9 + 576$
$\displaystyle \Rightarrow 6x = 585 - 225 = 360$
$\displaystyle \Rightarrow x = 60$
Now, what do you think the length of the ladder is?