Here are a couple of examples to show how these similarities can be used:

EXAMPLE 1:

Find x:

SOLUTION:   x is the long leg of the whole triangle and the hypotenuse of the triangle on the right.  So we use the ratio  for both.  Notice that the length of the hypotenuse of the whole triangle is 14 + 18 = 32.  Thus, .  Cross-multiplying gives .  Taking the square root, .

EXAMPLE 2:

Find x:

SOLUTION:  

Since x is part of the triangle on the right only and there are no given parts of this triangle, it is easier to find the hypotenuse of the whole triangle, and then subtract 3 to get x.  Thus we find y first:

Considering the whole triangle and the triangle on the left, we can use the ratio  for both:  .  Cross-multiplying gives , so , and therefore .