Problem of the Week: Cutting Up
Editor’s Note: The Department of Mathematics at Gettysburg College hosts a problem of the week challenge to determine each semester’s Paul Mugabi problem-solving award recipient(s). Each week’s entries are scored by a panel of faculty judges, and winner(s) from each week will receive a Problem Of the Week (P.O.W.) button. The Gettysburgian is not involved in or responsible for accepting or evaluating students’ submissions to this contest.
The contest is open to all Gettysburg College students. Up to three people may work together on a submission. Make sure your name is on your submission and that any sources are properly cited! Send solutions to firstname.lastname@example.org or put solutions in the marked envelope in the hallway outside Glatfelter 215. This problem was posted on Friday, September 7 and solutions are due on Friday September 14 by 5:00 p.m.
Recall that a polygon is convex if any two points inside it can be connected by a straight line that lies entirely inside the polygon (the polygon has no “indentations”).
If C is a convex polygon, drawing a straight line through it cuts C into two new polygons A and B, as in the figure below.
Can you find a convex polygon C (not necessarily regular!) and a straight line cutting through C so that one of the two new polygons created by the cut (let’s call it A) has the following properties?
- The area of A is precisely half the area of C.
- The perimeter of A is precisely half the perimeter of C.
[Hint: form C by putting a rectangle and a triangle together.]