Explorations in Elementary Geometry

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

The items below were inspired by various drawings, questions, or exercises I have seen recently. The answers to all of them may very well be known but they illustrate how one can try to personalize the generation of new problems - problem posing. In many ways problem posing is more important than problem solving. Different people are inspired to find mathematics interesting by different problems.

1. If one has two congruent rhombuses, what pattern of shapes can one obtain by placing the two rhombuses on top of each other?

Here I am assuming that you are given the rhombuses (thought of as including their interiors) with their edge lengths fixed. However, one might want to see what variation in results one can have if one has the right to choose the two congruent rhombuses that are used. For example, several "interesting" specific rhombuses are:

a. square

b. angles of the rhombus are 60 and 120 degrees

c. angles of the rhombus are 30 and 150 degrees

d. angles of the rhombus are 45 and 135 degrees

i. More specifically, one can ask for which values of k can the two rhombuses be placed so that their intersection is a k-gon?

ii. If the intersection of the two rhombuses is a k-gon, can this k-gon be:

a. equilateral?

b. equiangular?

c. regular?

iii. What is the largest area k-gon you can get for fixed k?

iv. What is the largest perimeter k-gon you can get for fixed k?

2. Above we assumed that we started with two congruent rhombuses. Do the answers to the questions raised above differ if the rhombuses are not congruent but have the same edge length? different edge lengths?

3. Repeat the investigation above where one uses other quadrilaterals than the rhombus. For example, you might look at the case of rectangles or kites.

Another direction to generalize this exploration would be to look at what happens for the intersection of a convex k-gon with a convex l-gon.