# Quadrilateral Puzzle

October 17, 2013I’ve been immersed in puzzle and lesson creation lately, and I thought I should take advantage and throw some of them out here on the blog. Please take, solve, use in your classrooms or at home, and let me know what you think. If people like the puzzles, I’ll make a point of putting them out here more often.

A Quadrilateral Question for today. This sub-questions goes from easier to harder.

**The Big Question**: Start with any quadrilateral (Quad 1), label its midpoints, and connect them to form another quadrilateral (Quad 2). When will Quad 2 take up exactly half the area of Quad 1?

Will it happen if Quad 1 is…

1. a square?

2. a rectangle?

3. a parallelogram?

4. a trapezoid?

5. a kite?

6. Nonconvex?

7. Can you find an example when Quad 1’s area isn’t double Quad 2’s? Or will it happen all the time?

You can post in the comments if you’ve got an argument to share…

(Another question is to show that Quad 2 is always a parallelogram. Here’s my proof of that if you get frustrated.)