Height of Intersection

Date: 01/16/2003 at 21:24:11
From: Rick
Subject: Advanced geometry

There are two flagpoles with heights of 10 feet and 70 feet 
respectively that are 100 feet apart on a level surface. If a line  
ran from the top of each pole to the bottom of the other, what would 
be the height of intersection?

Create a formula using one variable

I've concluded through diagrams that the distance apart has nothing 
to do with the location of the intersection because both the lines 
are moving proportionately, so the variable must be height.

Date: 01/16/2003 at 23:31:59
From: Doctor Peterson
Subject: Re: Advanced geometry

Hi, Rick.

You're right that everything depends not on the actual distance, but 
simply on proportions. What if we ignore the 100 in the problem, and 
just use variables for the two horizontal distances and the height of 
the intersection? (I don't see what the "formula using one variable" 
might be; but perhaps if we start with three variables, we'll see 
things simplify.)

Here's the setup:

    +
    | \
   a|   \     +
    |     X   |b
    |  / h| \ |
    +-----+---+
       x    y

Here a and b are the two heights, 70 and 10 in the original problem; 
I like to use variables (in this case, named constants) rather than 
numbers in problems like this where the specific numbers could easily 
be replaced with others, because that can give me more insight into 
the nature of the problem. The height we are looking for is h, and x 
and y are the horizontal distances.

Now I see two pairs of similar triangles. I'll set up two proportions, 
one relating a to h, and the other relating b to h. That gives two 
equations, which can then be solved to eliminate x or y. It turns out 
that both x and y can disappear; if you don't find that happening 
naturally as you solve the equations, try using each equation 
separately to get an expression for x/y in terms of a, b, and h, then 
set those equal. This will give you an equation you can solve for h; 
in fact, that might be the "formula" you are supposed to find. And 
it's the fact that only the ratio x/y matters, not the sum x+y, that 
your insight revealed.

If you have any further questions, feel free to write back.

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