Sections 13-5 and 13-6: Tangents to Circles and Uniqueness (Day 106)

Last night I tutored my student in geometry. He has now begun Chapter 5 of the Glencoe text, which is on Relationships in Triangles. Here are the sections of this chapter:

Section 5.1 -- Bisectors, Medians, and Altitudes of Triangles
Section 5.2 -- Inequalities in One Triangle
Section 5.3 -- Indirect Proof
Section 5.4 -- The Triangle Inequality
Section 5.5 -- Inequalities Involving Two Triangles

Let me point out the correspondences between the Glencoe and U of Chicago lessons. Glencoe's Section 5.1 is spread throughout several U of Chicago sections, but it is most noticeably appears in the same numbered section in the U of Chicago. But the other Glencoe sections appear nowhere near Chapter 5 in U of Chicago. The Triangle Inequality appears in Section 1-9 of the U of Chicago, and here "Inequalities Involving Two Triangles" corresponds to the SAS Inequality, Section 7-8 of the U of Chicago. (The Glencoe text also mentions a sort of converse to SAS Inequality, which it calls the SSS Inequality.)

But jumping out at us are Glencoe's Sections 5.2 and 5.3, for these correspond to sections in the chapter that we're covering here on the blog right now, Chapter 13. Section 5.2 in Glencoe refers to the Unequal Sides and Angles Theorems of Section 13-7 of the U of Chicago, and of course Indirect Proof appears in yesterday's lesson, Section 13-4 of the U of Chicago. This marks the first time that I tutored my geometry student on the same topic that is currently on the blog -- and in fact I showed him the very worksheet on Indirect Proof that I had posted just hours before!

My student, meanwhile, is having lots of trouble in this chapter -- so much that this week, he wants me to tutor him three times, yesterday, today, and tomorrow. And so I want to rearrange the lessons on this blog in order to match what he is struggling with, so that I can prepare extra worksheets specifically for him, as they would fit both Chapter 5 in Glencoe and Chapter 13 in U of Chicago.

First of all, he is definitely having trouble with Section 5.1. There are a few exercises in Glencoe where three points on the coordinate plane are given -- the vertices of a triangle. Then the student is asked to find the circumcenter, centroid, and orthocenter of the triangle.

Yes, I just mentioned the circumcenter in yesterday's post, except that I was discussing it in the context of an indirect proof that the perpendicular bisectors must intersect -- not on the coordinate plane, even though we could use the coordinate plane now that we've finally finished Chapter 11.

Imagine what finding the circumcenter of triangle ABC entails. If we are given the coordinates of the three points (A, B, and C), we begin by taking two of the points -- say A and B -- and now we must find the perpendicular bisector of AB. The "bisector" part is easy since we have already covered the Midpoint Formula. For the "perpendicular" part, we must find the slope of AB and then take its opposite reciprocal in order to find the slope of a perpendicular line. Now we have a point and a slope, so we have enough information to find the equation of the line. Afterwards, we repeat the process, taking another pair of points, say B and C, to find the perpendicular bisector of BC written for us as a second equation. Finally, we calculate where the two lines intersect -- we don't need the third because we know that they are concurrent -- by solving the system of equations.

As you can see, this is a lot of work. It requires a lot of Algebra I -- we briefly mentioned some of this in U of Chicago Chapter 11, but the bulk of it is from the student's previous course. Many students aren't necessary comfortable with this -- for example, many students don't remember the Point-Slope Formula, and so I must either remind them or use the Slope-Intercept Form. Similarly, I must remind the students how to solve a system of linear equations by substitution or elimination.

And then, all of that is just to find the circumcenter! The Glencoe text, in one of the exercises, also directs the students to find the centroid and orthocenter! Finding the orthocenter is also like finding the perpendicular bisector, except that this time, we find an equation through the opposite vertex, with the calculated perpendicular slope.

Finding the centroid is easier. This is because we can simply average the coordinates -- recall the concept of center of gravity mentioned back in Section 11-4. For a polygon of four or more vertices, the center of gravity of the whole region is not the center of gravity of the vertices -- but it's implied that for a triangle, they are. (If the density of the material is constant, then "centroid" and "center of gravity" mean exactly the same thing!)

And if we just choose random points on the plane, chances are good that the coordinates will have fractions with large denominators. (I remember the denominator 38 from last night!) The coordinates of A, B, and C could be selected so that the circumcenter would be a lattice point (that is, one whose coordinates are integers), but then the centroid and orthocenter will still be nasty. Also, one could choose A, B, and C so that two of them lie on a horizontal or vertical line -- this would make finding the equation of the perpendicular much simpler. But of course, the Glencoe text does no such thing.

Even though I wrote that I would include more algebraic problems during second semester, this type of problem is well out of control. (Moreover, recall that we're still only in the fifth chapter of the Glencoe text, so this is still the first semester of the text!)

I wanted to come up with some triangles for which the three centers don't come out nasty. Notice that to avoid fractions when finding the midpoints, I want even coordinates, and to avoid fractions when finding the centroids, I want multiples of three. And so I choose multiples of six. For example, I chose the coordinates A(18, 0), B(0, 0), and C(12, 12). For that example, one of the sides is horizontal to make it easier. For my second example -- one where I don't have a horizontal or vertical side -- I chose the values A(20, 10), B(0, 0), and C(16, 16). Since these coordinates aren't multiples of three, there's still a fraction in the y-value of the centroid. But this is much better than having a fraction in an intermediate step (for example, a fraction in the y-value of the circumcenter when one still needs to plug it into an equation to find x). This is what will intimidate the student.

And so I really want to create some worksheets to help him out. It's a shame that I already wasted time on today's posted worksheet for Section 13-5 of the U of Chicago text -- tangents of circles has absolutely nothing to do with my student's lesson. My plan is to see what in Glencoe's Chapter 5 my student is still having trouble with, and then base tomorrow's worksheet on that. But I have absolutely no time to try to get another worksheet ready for tonight.