Construct a Trapezoid

Date: 08/28/2001 at 13:24:35
From: Sarah
Subject: Geometry Construction

There are four line segments: b = 2cm, c = 3cm, d = 4cm, and f = 6cm.  

Problem: To construct (with straightedge and compass) a trapezoid 
having b and f as bases and c and d as legs.

I tried drawing two lines that are parallel to each other for b and 
f, and I drew c, but then d didn't fit. How do I construct this?

Date: 08/28/2001 at 14:24:59
From: Doctor Peterson
Subject: Re: Geometry Construction

Hi, Sarah.

The problem with your approach is that you assumed the parallel sides 
were a certain distance apart, but the sides determine this distance, 
so you don't have room to choose it for yourself. The hard part is 
that it takes all four lengths to determine the height, so you can't 
construct the correct height easily to start with.

Here's a nice trick for working with trapezoids:

             b
        +---------+                +
       /|         | \             /| \
     c/ |         |   \d   ===> c/ |   \d
     /  |         |     \       /  |     \
    +---+---------+-------+    +---+-------+
             f                      f-b

If you cut out the rectangle from the middle, you are left with a 
triangle. (It will be a little different if the top is not directly 
over the base, but the construction should still work.)

You know all four sides, so you can find all three sides of the 
triangle. You probably know how to construct a triangle given its 
sides; so do that, and then put the rectangle back in to make the 
trapezoid. This drawing might be helpful for that last step:

                              b
        +                +---------+
       /  \             /  \         \
     c/     \d   ===> c/     \d        \d
     /        \       /        \         \
    +-----------+    +-----------+---------+
         f-b              f-b         b

You'll probably have to refine these ideas a bit to make a neat 
construction, but this should give you a good start.

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