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### 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/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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