N Equal Parts by Construction

By Arthur N. DiVito, Ph.D. (copyright © 2000 by A. N. DiVito)



Straightedge (i.e., unmarked ruler) and compass constructions are a hallmark of geometry courses. Students really should do in class such classical constructions as bisecting an angle, forming a right angle, and drawing a line parallel to a given line through a given point off the line. However, one of the most important facts in all of mathematics is that a given line segment can be subdivided into any prescribed whole number of equal parts. For example, 5/19 “makes sense” because the unit length can be divided into 19 equal sublengths, and the first 5 can then be identified. One ramification of this is that the position of any rational number, upon the number line, can be found by straightedge and compass construction. [Note: One must not confuse the interesting question of whether it is possible “to divide a given length into, say, 5 equal parts” with the easy matter of “producing a line segment which is marked off into 5 equal parts.”]


The activities of this lab are to have teams of students determine, say, precisely 1/5 of a given straight line segment, using straightedge and compass construction; thus establishing that the segment can be divided into 5 equal parts. Of course, any whole number can be used. 5 is used in this description, but the students should be assigned various numbers of equal parts. There are two excellent methods for doing this. The first is the classical construction which was known to the ancients. The second appears-remarkably-to be an original idea which was discovered by a couple of high school students relatively recently.

Classical construction: The segment to be divided is placed horizontally. At, say, the left endpoint of the segment, a line divided into exactly 5 equal parts is drawn (it could be placed perpendicularly, but this is not necessary). The unconnected endpoints of the two lines are now connected with a straight line. Finally, lines parallel to this line are constructed through the remaining four dividing marks of the second line. These lines will intersect the horizontal segment in such manner as to partition it into five equal parts.

Modern construction: The segment to be divided is placed horizontally. A rectangle is constructed (of any height) using this segment as width. The diagonals of the rectangle are constructed. A perpendicular dropped from the point of intersection to the original segment will divide the segment into two equal parts at a point, say A. Now draw a line from the upper left corner to the point A. A perpendicular dropped from the new point of intersection will provide a point, say B, one-third the way from the left endpoint. Continue in this manner to produce one-fourth, etc.


Empirical verification is just a simple matter of measuring (or using the compass).


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