Height of a Trapezoid

Date: 02/08/2002 at 19:30:35
From: Sam 
Subject: Formula for the height of a trapezoid

A trapezoid has parallel bases of lengths 5 and 30, and non-parallel 
sides of length 5 and 25.  Find the height of the trapezoid.

Date: 02/08/2002 at 23:45:36
From: Doctor Peterson
Subject: Re: Formula for the height of a trapezoid

Hi, Sam.

Look at the trapezoid:

      D   5   C
       +-----+
    5 /:     :  \  25
     / :     :     \
    +--+-----+--------+
   A        30         B

I've drawn in a rectangle that I want to cut out of the trapezoid. 
I'll glue the remaining triangles together; that will leave me with a 
triangle whose sides I know:

       C
       +
    5 /:  \  25
     / :     \
    +--+--------+
   A     25      B

What's interesting here is that this turns out to be an isosceles 
triangle. There are several ways to work out the altitude of this 
triangle; here's my approach:

        C
        +
    5  /| \  25
     F+ |h  \
     /  |  \  \
    +---+-------+
   A    E 25     B

Draw an altitude from C to AB at E, and another from B to AC at F, 
dividing ABC into two congruent right triangles. Then triangle AEC is 
similar to triangle AFB, and altitude BF is sqrt(25^2 - 2.5^2). Then

    CE   BF       h    sqrt(25^2 - 2.5^2)
    -- = --  so  --- = ------------------
    AC   AB       5           25

This gives

    h = sqrt[(25^2 - 2.5^2)/25] = sqrt[25 - .25] = sqrt[100 - 1]/2
      = 3/2 sqrt(11)

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