Chord Star: Another Geometry-Function-Modeling Thing

Last time I wrote about a super-simple geometry situation and how we could turn it into an activity that connected it to linear functions. What does it take to turn something from geometry into a function? This is an interesting question; in my explorations here I’ve found it helpful to look for relationships. And what I mean by that is, where do you have two quantities (in geometry, often distances, but it could be angles or areas or…) where one varies when you change the other.

So one strategy is, think of some theorem or principle, and see if you can find the relationship. To that end, remember teaching geometry and that cool theorem where if you have two chords that cross, the products are the same? That’s where this comes from. Oddly, it took a while to figure out what to plot against what to get a revealing function, but here we go.

Make a circle. Pick a point not near the center, but not too close to the circle itself. Draw a chord through that point. Measure the two segments. Call them and . Or even x and y. Record the data.

And repeat. Draw a different chord, record the two segment lengths. And again. You’ll soon have a “chord star” and a bunch of ordered pairs. Since you can’t really see it in the top illustration, here’s a (slightly blurry, sorry) detail:

Now plot. And fit. I used Desmos and the function where k is a slider, but that’s because I had an idea how it should turn out. Students might try any number of things.

At right, the finished plot. If you want to play with my actual data on Desmos, here is the link. But of course you should really trace a saucer, make a point, get a ruler, and take data yourself!

A few notes

• The astute reader may realize that the two points are interchangeable, so there can actually be twice as many points as I plotted.
• An interesting question is, “What effect does your choice of point have on the graph and the data?” More on this next time.
• If you didn’t know this theorem, you could look at the relationship and say, “Interesting! The product of any two partial-chords is a constant!” Could that give you insight into the geometrical situation? Could it help you with the proof?
• What does this approach get you that may be just as good (or better) than the synthetic-geometry approach you take in a formal geometry class? Is there a benefit in having both? (Sure there is! At least one idea about this to come…)
• This is an attempt not to spend 3 hours making a blog post, which has sort of been my minimum, and a big barrier. So I hope this tells enough even though it doesn’t have carefully-crafted reflection all over it.