Saturday, July 20, 2013

Good Problems: Geometric Construction

I enjoy the problems posted on the Five Triangles blog. It says they're appropriate for grades 6 to 8, but they are often hard enough to make me sweat a bit. This particular problem demanded to be solved, and made me sweat more than most.

When I finally got a chance to look it over carefully, my first response was bewilderment. Then I had a few ideas, got out paper and pen, and drew some lines in. I was stuck again, but when I stated my new simpler problem carefully, and drew a new picture for it, I got it. To check, I used geogebra to construct it with just points, lines, and circles.

Playing with these problems makes me want to teach a geometry course. 


  1. Is it cheating to use a corner of a piece of paper to do it? They both have to be 45 degrees right? So splitting the difference is easier using the right angle of a corner of paper and making both sides of the paper touch points A and B.

  2. Will that make both angles be 45 degrees? I think if it slides around, they could be different values. But I like your creative thinking.

    Also, the way the picture is drawn, your assumption seems reasonable, but I think the idea is to find a way to do it even if the angles aren't 45 degrees. If you imagine A and B coming straight down, so they're a lot closer to the line, then the angles won't look like 45 degrees. There's still a way to do it.

  3. Ah, ok, I found one correct answer visually, but now I need a formal answer.

  4. I know one technique, using vertical angles. I'm not sure if there's an easier way.

    As for the angles, here's an easy way to see that they don't have to be 45 degrees: imagine moving the two dots horizontally, either very close to each other or very far apart. In the first case, the angles would approach 90 degrees, while in the second case, the angles would get smaller the farther you moved the dots.

  5. As the posers of this problem, we can offer some background (but not hints...heh) about this construction, and constructions in general, summarized here:

  6. @Denise: I think the solution using vertical angles is the simplest.

    Initially, I had a more complicate solution using parallel lines and similar triangles (inspired by some of the construction challenges in the "game" Sue linked previously), but the vertical angles solution requires just 3 circles and 1 straight line, so I'm pretty convinced that it is the "simplest".


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