MEDIAN OF A TRAPEZOIDAuthor(s): Pamela AllisonSource: The Mathematics Teacher, Vol. 79, No. 2 (FEBRUARY 1986), pp. 103-104Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27964800 .

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ativity." Journal for Research in Mathematics Educa tion 5 (November 1974): 197-211.

A lice Artzt

Queens College of the

City University of New York

Flushing, NY 11367

MEDIAN OF A TRAPEZOID The geometric theorem that states that the median of a trapezoid is parallel to the bases and that its length is one-half the sum

of the lengths of its bases is found in vir

tually every first-year geometry text. It usu

ally follows the theorem stating that the

segment joining the midpoints of two sides of a triangle is parallel to, and one-half of, the measure of the third side. These two

theorems?the median theorem and the

triangle theorem?seem similar, and the

proofs presented by the texts do make use

of the triangle theorem to prove the median theorem (fig. 1). In fact, the proof of the median theorem is usually presented by

constructing a line through one vertex of the trapezoid and the midpoint of the op

posite leg and proving congruent triangles so that corresponding parts are congruent. In figure 2, AM2 is constructed and

AAM2 s AGM2 C so that AB s CG. AADG is used to complete the proof (Hirsch et al. 1979).

a y ?

D C

(b)

Fig. 1

Fig. 2

Many theorems can be proved in various

ways, and the median theorem is no excep tion. An alternate proof exists that does not

necessitate the use of congruent triangles (fig. 3). Start with trapezoid ABCD. Mi is the midpoint of leg AD and M2 is the mid

point of leg BC. Construct segment BD

with midpoint M. By the triangle theorem is parallel to S3. Likewise, mM2\\

CD. Therefore, by the transitivity of paral lel lines, both and MM2 are parallel to AB We know that through a point M not on there exists only one line paral lel to AB, so we can be certain that Ml9 M, and M2 are collinear. Now, by the triangle theorem, if AB = y, MMl = \y, and if

CD = jc, MM2 = i*. By adding, we complete our proof:

MM, + MM2 = ^ ( + y)

The length of the median is, therefore, one

half the sum of the lengths of the bases.

Fig. 3

One can easily see the similarity be tween the triangle theorem and the median theorem. In my classes I draw trapezoid ABCD and its diagonal in white chalk, then shade AABD and its interior in

orange chalk ?nd AC DB and its interior in blue. (The colors can be varied for the aes

February 1986 103

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thetes among you.) We then talk about the

"orange" triangle with its orange segment, MMly and the "blue" triangle and its corre

sponding segment. After the students are convinced that the length of the median is indeed half the sum of the two bases, I usu

ally draw a single triangle in orange next to our colorful trapezoid and have the stu dents give me similarities between the two theorems. With some intensive guidance, the students can usually come up with cor relations. In many ways, the trapezoid can be thought of as the triangular figure "dou bled":

The triangle's segment (orange, of

course) is parallel to, and one-half of, one

side, whereas the trapezoid's median is

parallel to, and half of, two sides.

Th? triangle with its midpoint segment contains two pairs of congruent correspond ing angles (fig. 1(a)). LRM^ ^ LRST and lRM2M1^lRTS. The trapezoid with its median (and no diagonal drawn) contains four such pairs.

The triangle contains four pairs of sup plementary angles, including both linear

pairs and interior angles on the same side of transversals and M2^. The trap ezoid and its median contains eight pairs of

supplementary angles.

It's unfortunate that the terminology is not also similar ; it would be nice if the seg ment joining the midpoints of two sides of a

triangle was called the "median of the triangle.

"

Since developing this method of proving the median theorem, I've discovered that some texts use it. I developed it originally because I could see its possibilities for

showing the relationship between the two theorems. The advantage to the diagonal proof is that it does show the trapezoid as two triangles, which leads nicely into the list of similarities.

Of course, whatever proof is used, the

relationships remain the same. If the two theorems are linked in the students' minds, they may become easier to recall. In any case, we now have an alternate method of

proving an interesting theorem, along with an excellent chance to use colored chalk.

REFERENCE

Hirsch, Christian R., Mary Ann Roberts, Dwight Co

blentz, Andrew Samide, and Harold Schoen. Geome

try. Glen view, 111. : Scott, Foresman & Co., 1979.

Pamela Allison East Bay High School Gibsonton, FL 33534

PLEASE CHECK YOUR TELEPHONE DIRECTORY The telephone directory holds a wealth of information usable in a mathematics class room. When students get tired of the usual textbook fare, why not use the telephone directory as a teaching aid? Multiple copies of outdated directories can usually be ob tained from the telephone company, or stu dents can bring in old copies from home. How can you use the directory in a math ematics class? Here are some suggestions for various areas of mathematics. You will need to decide the appropriate grade levels for your classes.

Probability

Suppose that you forgot the last digit of a

seven-digit telephone number. If you choose one of the digits from 0 to 9 as a guess, and ten correct digits are possible, is the prob ability of correctly guessing any digit in the last position the same? In the third position from the left?

An empirical experiment can be con ducted to estimate these probabilities. For the second question, for example, randomly choose a beginning number on a page of the

directory and look at the third digit from the left in the next 100 telephone numbers.

Make a frequency table for the digits from 0 to 9 and tally the number of times each occurs in the third position for these 100 numbers. (In a larger community with many telephone exchanges, you may need to tabu late the frequencies of the second digit.)

Table 1 can give you a good estimate for the missing third digit, since in most areas, some telephone exchanges occur more fre

quently than others. In some localities,

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?-Mathematics Teacher

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