## Chord Star 2: Choosing different points

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 $L_1$ and $L_2$) and plotted them against one another. We got a rectangular hyperbola, suggesting (or confirming if we remembered the geometry) that $L_1 L_2 = k$, 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.