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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 + y
a
 = 
 y 
h
 and 
x + y
b
 = 
 x 
h

Adding equations:

x + y
a
 + 
x + y
b
 = 
x + y
h

Dividing by (x + y):

 1 
 a
 + 
 1 
 b
 = 
 1 
 h
Problem ID: 127 (Oct 2003)     Difficulty: 3 Star

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