## Alley Ladders

#### Problem

Two ladders are placed on opposite diagonals in an alley such that one ladder reaches `a` units up one wall, the other ladder reaches `b` units up the opposite wall and they intersect `h` units above the ground.

Prove the following result.

1 a | + | 1 b | = | 1 h |

#### Solution

Consider the following diagram:

By similar triangles:

x+ya= yhand x+yb= xh

Adding equations:

x+ya+ x+yb= x+yh

Dividing by (`x` + `y`):

1a+ 1b= 1h

Problem ID: 127 (Oct 2003) Difficulty: 3 Star