Last time we saw how you could make a “chord star” by picking a point inside a circle and drawing chords through that point. Then we measured the two lengths of the partial chords (let’s call them and ) and plotted them against one another. We got a rectangular hyperbola, suggesting (or confirming if we remembered the geometry) that , some constant.

But we asked, “what effect does your choice of point have on the graph and the data?” So of course we’ll take an empirical approach and try it. If you have a classroom full of students, and they used the same-sized circle and picked their own points, you could immediately compare the points they chose to the functions they generated. Or you could do it as an individual. The photo shows what this might look like, and here is a detail:

Now we’ll put the data in a table, but this time,

- In addition to L1 and L2, we’ll record
*R*, the distance from the center to the point. It may not be obvious to students at first that all points the same distance from the center (or the edge) will give the same data, but I’ll assume we get that. - We’ll double the data by recording the data in the reverse order as well. It makes the graph look better.

Here’s the graph, coded by distance (in cm) of the point from the center.

The *R* = 7 points (the “dels”) look as if they might lie along a hyperbola; the others turn out to do so too. Without zooming in and showing what we hope students would do (e.g., look at residual plots), let’s skip ahead and plot these functions on the graph. Aha: part of the answer to our question is, if you choose a different point, the relationship between and has the same *form*, but the points lie along a different hyperbola.

Well. There’s clearly a systematic relationship between the position (radius) of the point and the curve it generates. What is that relationship?

This gets back to the question, what can we plot? (*We* know, but how can we help students make this decision for themselves?) One intriguing idea is to make a 3-D plot, so we would see each hyperbola at an *altitude* (*z*-value) given by that radius number. Great idea! I don’t feel like wrestling with the appropriate tools, though, so I’ll ask, “is there any way we can characterize the function with a single number?”

Heck yeah. The functions are, for example, for *r* = 7; for *r* = 5.5; for *r* = 4, and so forth. They’re all just . Let’s just use the value of *k*.

Here is the graph of *k* against *r*.

Okay, reader, what function would you use to model this? And why does it work? (Don’t tell, of course! Just work it out.)

This is not the only “next-level” relationship that you can pull out of this situation. If you look back at the first data graph, we have to ask, what points are possible? It looks as if there’s a linear boundary, a negative diagonal, separating the allowed area from the forbidden. Is that right? If so, what’s the line, and why?

Additional questions (yes, you want students to ask these, but I’m putting them here so I don’t forget them. The best way to answer most of them, by the way, is to understand what they mean *geometrically*—they’re about chords and circles, after all, not hyperbolas.):

- How come the hyperbolas can go into the forbidden zone?
- Are all points
*below*the chord-star boundary possible? - If you have a single point from a graph, what can you tell about the chord it came from?
- If you have a single point, what can you tell about the circle it came from?
- If you have a single hyperbola, what can you tell about the circle?
- What’s the least amount of information you need from a table of data like ours to determine the circle?
- How can we take stuff like this out into the real world? Stay tuned for our next exciting episode.

The forbidden-zone issue smells a lot like the “random rectangles” plot you can make, plotting area against perimeter (as I did long ago in *The Mathematics Teacher*, vol 94, no 8, November 2001). At right is a graph from that article showing areas and perimeters for 100 randomly-generated rectangles with lengths and width in the range zero to 30. There’s the curvey boundary, but also a straight, slanty one. See if you can understand both.

Finally, for lazy persons who don’t want to measure their own data, but are still interested enough to play with it, my measurements in centimeters, tab-delimited, I hope:

L1 L2 R 8.6 7.8 1.0 8.8 7.2 2.5 7.8 8.6 1.0 7.2 8.8 2.5 4.6 11.5 4.0 11.5 4.6 4.0 6.3 8.4 4.0 8.4 6.3 4.0 5.9 6.6 5.5 6.6 5.9 5.5 11.4 3.5 5.5 3.5 11.4 5.5 4.5 4.4 7.0 9.5 2.1 7.0 2.1 9.5 7.0 9.3 7.3 1.0 7.3 9.3 1.0 10.8 5.8 2.5 5.8 10.8 2.5 15.3 1.3 7.0 1.3 15.3 7.0 2.8 13.8 5.5 13.8 2.8 5.5

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Tim

Thanks for two fantastically helpful posts. I have been doing a great deal of thinking about our Geometry curriculum and these are the kinds of activities I want our kids engaged with. Another thought regarding your question about hyperbola graphs going into the ‘forbidden zone’ – I think that this ties in well with algebra discussions of extraneous solutions and with the recognition that while equations can model situations the solution sets are not always identical.

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