So You Think You Can Find a Midpoint?

There may well have been a time when learning how to do geometric constructions with only a compass and straightedge was a valuable practical skill. But if there was a time when those were the tools used in professional drawing, that time is centuries gone.  Now we have fancy computers.  But we still learn about these constructions in geometry class, and that’s not a bad thing — that kind of problem-solving (I would say puzzle-solving) is how we learn and discover.  Since we do constructions to understand abstract geometric relationship, not to become compass-and-straightedge whizzes, there’s no reason to limit ourselves to those two construction tools!

At the spring meeting of the Michigan section of the Mathematics Association of America, I had the good fortune to meet Tibor Marcinek of Central Michigan University.  He talked about educational uses of a set of nine java applets concerned with finding midpoints of a line segment using various sets of tools, and challenged us to solve them.  Probably the most fun I had at that whole conference.

You can find the challenges here.

Pedagogical value aside, I think they work as geometry-based puzzles for anyone to tackle.

WARNINGS

  1. You can’t just eyeball it. The program knows whether you found the midpoint by a construction that works in general or you are just guessing.  If you’re right, it will tell you.
  2. Some of the puzzles are a lot trickier than they first appear.
  3. You might accidentally learn some geometry facts.  (But that can be our secret.)

If you get more than half of them, please post a gloating comment and get your due.

It might not be obvious at first what all the tools do.  If you’re the sort who likes to figure everything out for yourself, stop reading now and go play.

Puzzle 1.

This is the classic construction with a compass and straightedge.  You probably learned how to do this in high school, and you probably forgot.

  • Point. Mark a point on the screen (such as the interesction of two lines or circles). If you mark the midpoint, you win!
  • Straightedge. Draw the line through two points.
  • Collapsible compass. Draw the circle centered at a point and through another point. (It’s called a collapsible compass because it collapses when you pick it up; you can’t measure off a distance and then draw a circle with that radius centered somewhere else.)

Puzzle 2.

  • Point.
  • 9-point line. Draws a line with nine equally-spaced points, including two points you choose.
  • Parallel. Draw a line through a given point parallel to a given line.

Puzzle 3.

  • Point.
  • Segment Straightedge. Draw the line segment through two points.
  • Translation. Draw an exact copy of a given line segment starting at some third point.

Puzzle 4.

  • Point.
  • Segment Straightedge.
  • Parallelogram. Given three points, draw a parallelogram whose vertices include those points.

Puzzle 5.

This one’s tricky. A lot of things that seem like they ought to work don’t.  Might be wise to try the next couple and come back to it.

  • Point.
  • Straightedge.
  • Half-distance. Given two points A and B, mark a point on that line, but on the far side of B, half as far from B as A is.

Puzzle 6.

  • Point.
  • Straightedge.
  • Triangle-by-centroid. Draw a triangle with two given vertices and a given center of mass.

Puzzle 7.

  • Point.
  • Straightedge.
  • Triangle with 2 midpoints. Draw a triangle with one side as the starting line and mark the midpoints of the other two sides.

Puzzle 8.

  • Point.
  • Straightedge.
  • Midsegment parallelogram. Draw a triangle with one side as the starting line and draw the parallelogram whose center is the third vertex and one of whose sides connects the midpoints of the other two sides.

Puzzle 9.

At first this one seems impossible, but it’s really satisfying when it finally clicks!

  • Point.
  • Collapsible compass.
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3 Responses to So You Think You Can Find a Midpoint?

  1. […] Permalink: So You Think You Can Find a Midpoint? […]

  2. Ibrew says:

    Those weren’t too hard, but i might have given up on the last one if i hadn’t assumed it was possible like the others. Finding out what the tools do (and don’t do) was the hard part.

    And…Wow. These posts are great! I found them trying to make sure i was calculating odds correctly on the Second Life game Greedy Greedy (Farkle, with some minor rule tweeks). Oh, and i wasn’t. Thanks again 🙂

  3. Nick says:

    I guess would have struggled with the last had I not known before hand about the Mohr-Mascheroni Theorem (that all euclidean constructions can be accomplished with collapsible compass alone). I’ve tried for a good half hour on the fifth though, and still can’t nut it out, I’m just about ready to give up. 😦

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