Projects - Cengage Learning

Appendix C C-51
PROJECT
Constructing a Regular Pentagon
This is intended as a group project to create a poster explaining a compass and
straightedge construction of a regular pentagon. (Recall that regular means
that all of the sides are equal and all of the angles are equal.) In the first part
you are going to construct a regular pentagon by first constructing a regular
decagon and then connecting every other vertex. The second part asks you to
write a short paper explaining why the construction is valid. Although the construction
given here is somewhat awkward, its justification is fairly straightforward.
In the Writing Mathematics section at the end of Chapter 9* a shorter
and more elegant construction is described. The third part provides some history
of the construction of regular n-gons.
(a) As background you should review the ruler and compass construction for
the perpendicular bisector of a line segment. The construction proceeds as
follows.
Draw a line segment, and label the endpoints A and D. Construct the
perpendicular bisector, and denote the midpoint of AD by C. Set your
compass to the length of AC, which is 1 unit for this construction,
and mark a point two units from C along the perpendicular bisector.
Label this point B, and draw the line segment AB. With your compass
still set to the length of AC, draw a unit circle somewhere away from
triangle ABC. Set the needle tip of your compass at B, and let E denote
the point on AB that is 1 unit from B. Bisect the line segment AE,
and label the midpoint F. The length of AF will be one side of a regular
decagon inscribed in the unit circle. Mark ten consecutive points
around the unit circle spaced this distance apart. These are the vertices
of a regular decagon. Connecting every other vertex completes
this construction of a regular pentagon.
(b) As background for this part you should review parts (a) though (f) of
Exercise 54 in Section 7.5*. To complete this part of the project, write a
careful justification for the construction described in part (a). You should
use a mixture of English sentences, equations, and figures similar to the exposition
in this textbook. A brief sketch of the beginning of a justification
follows:
It is straightforward to show that the line segment AF described in
part (a) has length 115 122. From Exercise 54(f) in Section 7.5,*
sin 18° 115 124. This implies, in a few steps, that an isosceles
triangle with two sides of length 1 and the third side of length
115 122 has a vertex angle of 36°.
(c) Carl Friedrich Gauss (1777–1855) was one of the greatest mathematicians
in history. In the last section of his great work on number theory,
Disquisitiones Arithmeticae, published in 1801, he discusses the problem
of constructing regular n-gons. He first deals with regular polygons with a
prime number, p, of sides and proves that if p is of the form p 2 2k
1,
where k is a nonnegative integer, then a regular p-gon is constructible, that
*Precalculus: A Problems-Oriented Approach, 7th edition