¶ 1994 AHSME Problem 26
Problem:
A regular polygon of sides is exactly enclosed (no overlaps, no gaps) by regular polygons of sides each. (Shown here for .) If , what is the value of
.jpg)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The measure of each interior angle of a regular -gon is . In this problem, each vertex of the -gon is surrounded by one angle of the -gon and two angles of the -gons. Therefore,
and gives .
Note. The equation may be written in the form . Its only solutions in positive integers are , and .
Each interior angle of a regular decagon measures . The interior angles of the two -gons at one of its vertices must fill . The regular polygon each of whose interior angles measures is the pentagon, so .