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.

4 comments:

  1. I made an equation in two variables, and asked wolfram alpha to give me integer solutions. The one they gave me made me laugh. Turns out my joy and yours are probably unrelated. Even with the added line, this problem doesn't look boring to me. I'll have to keep thinking about geometric solutions.

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    1. For me the numerical solution came quickly if I assumed its area is constant however we change the unknown sides. I just made the top unknown segment zero, essentially dragging the red dot to the left mentally (I hadn't built the applet then).

      The more interesting question to me was why was the area constant. The question wouldn't make sense if it changed, but why? For some reason, I didn't make the connection earlier even though I found the area by mentally drawing that segment so I could calculate the base of the triangle in he bottom left.

      Perhaps we both found joy in the question because the problem was easier than it seemed at first. It's "how didn't I know that I knew it" kind of happy realization.

      May be I should've used common or straightforward instead of boring.

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  2. The problem was originally posted here:
    http://fivetriangles.blogspot.com/2012/04/area-problem.html

    And here was some discussion about it on Reddit:
    http://www.reddit.com/r/math/comments/1c46c7/does_this_quadrilateral_area_problem_really/

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    Replies
    1. Thanks for the links. I like the other area problem too. Two great questions about the same concept.

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