Wednesday, July 17, 2013

Fun Little Area of Quadrilateral Question

Fun little problem about the area of a quadrilateral tweeted by @daveinstpaul. Try it out.

If you're stuck, try the following applet and click on the checkboxes labeled "Show Hint ##"

mrhodotnet, 17 July 2013, Created with GeoGebra

Some questions that popped in my mind as I worked on this question:
  1. Is the shaded quadrilateral unique? Is this the only quadrilateral that fits the criteria (given sides and angles)?
  2. Is it possible to draw another diagram with the same given sides and angles but with different shape?
  3. Isn't it the case that given only 4 sides of a quadrilateral, there isn't enough information to determine area? If so there may not be enough information to determine area, but if it were the case the question would be pointless.
  4. Is it possible that the area of the shaded quadrilateral is constant regardless of the shape?
  5. If the area is constant, let's choose the diagram that is the most calculation friendly and find the area.
  6. I have the area now. How do I know it's constant? Can I show this? What principles are at work? What can I learn about the special cases (the diagrams that I drew) to show it for the general case?
  7. Is it possible to show this using trigonometry? How should I break down the quadrilateral? A trapezoid and a triangle? Two triangles? Should I tackle the unshaded area first?
  8. Let's label the unknown sides and use Pythagorean Theorem to find the relationship between the missing sides. Do the equations simplify?
It was some time later, while doing chores, when it hit me. There is a kind of joy and excitement when an idea strikes and it's immediately clear that it'll work. A simple auxiliary line segment is all it was, probably immediately obvious to Geometry teachers. It's interesting how this question was one auxiliary line segment away from being boring. Glad it was omitted from the original.

This is my first time embedding a tweet in a post and embedding a GeoGebra html5 applet instead of a java applet. People really make these things easy to do nowadays. Thanks Twitter and GeoGebra!

Update: Removed tweet embedding until I figure out how to embed without including media.