Question

How far from the base of the house do you need to place a 15ft ladder so that it exactly reaches the top of a 12ft tall wall

Answer 1

It's a right triangle problem, and pretty obviously a multiple of the only Pythagorean Triple most folks know, 3²+4²=5².  In this case we multiply the lengths by three getting

9²+12²=15²

If we place the 15 foot ladder 9 feet from the house it will hit the house at height 12 feet.

Answer: 9 feet