Similar Triangles and Ratios

Date: 12/04/2002 at 20:43:20
From: Aida
Subject: Math problem for a math test 

I tried but I just don't get it.  My question is:

A man who is 54.4 inches tall casts a shadow that is 69.7 inches. His 
son's shadow is 41 inches. What is the height of the man's son ?

Date: 12/05/2002 at 12:02:26
From: Doctor Ian
Subject: Re: Math problem for a math test 

Hi Aida,

Here is a picture of the situation:

       | .
  54.4 |   .        | .
       |     .    ? |   .
       |_______     |_____   
         69.7         41


The key principle here is that the triangles are 'similar' to each
other, which means that even though they are different _sizes_, the
ratio of any two sides in one triangle will be the same as the
corresponding ratio in the other triangle. In this case, it means that

  54.5     ?
  ---- = ----
  69.7    41

Two triangles are similar whenever all of their angles are identical.

Imagine drawing a triangle on a sheet of paper, and then making a
photocopy with the zoom set to something other than 100%. The lengths
of the lines will change by the same amount - e.g., if one side is
doubled, the other sides will be doubled as well - but the angles will
remain unchanged. In other words, if you scale the size of a triangle, 
you end up with a triangle that is similar to the original triangle.

And that's kind of what's happening here. The triangle for the man is
a scaled ('zoomed') copy of the triangle for his son.  

Can you take it from here? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/