Friday, May 12, 2017

Lesson 15-3: The Inscribed Angle Theorem (Day 153)

This is what Theoni Pappas writes on page 132 of her Magic of Mathematics:

"Select a continent. Let's use South America as an example. Suppose we want to measure the distance of its coastline."

This section is called "Fractalizing the Surface of the Earth." Hey -- we're still talking about fractals, so let's continue with what I wrote about Mandelbrot last year:

Part III of Mandelbrot's The Fractal Geometry of Nature is called "Galaxies and Eddies," and consists of Chapters 9 through 11. In this chapter, Mandelbrot argues that many problems that appear in science can be solved by using fractal geometry.

For example, in Chapter 9, Mandelbrot discusses the distribution of stars in the universe. He writes about Olbers paradox, which poses the question, if the universe is uniformly full of stars -- that is, if there are stars everywhere -- then why is the sky dark at night? Mandelbrot -- who prefers to call this the "Blazing Sky Effect" rather than Olbers paradox -- proposes that it could be because the universe is not uniformly full of stars. Instead, the universe could be a fractal with dimension less than 3, and so the distribution of stars follows a fractal pattern.

In the ensuing chapters, Mandelbrot mentions other scientific ideas -- turbulence and fluid motion -- that could also be modeled using fractals. So we see that these fractals are hardly some abstract mathematical idea that has no practical significance, but instead have some scientific value.

Returning to 2017, let me say that Mandelbrot does write about coastlines in his Part VII. So coastlines are another scientific problem that is solved using fractal geometry. Pappas writes about how the length of a coastline seems to increase when measured with a shorter ruler. The reason is that the coastline doesn't have a finite one-dimensional length. The dimension of the coastline is between 1 and 2 somewhere.

This is what I wrote last year about today's lesson:

Lesson 15-3 of the U of Chicago text is on the Inscribed Angle Theorem. I admit that I often have trouble remembering all of the circle theorems myself, but this one is the most important:

Inscribed Angle Theorem:
In a circle, the measure of an inscribed angle is one-half the measure of its intercepted arc.

The text divides the proof into three cases -- depending on whether the center of the circle is inside, outside, or on the inscribed angle. The easiest case occurs when the center is on the angle, and this case is used to prove the other two cases. The exact same thing occurred when we proved the Triangle Area Formula back in Chapter 8. Then, the three cases were whether the altitude was inside, outside, or aside of the triangle -- and the case when the altitude was aside of the triangle (i.e., when the triangle was a right triangle) was used to prove the other two.

I will reproduce the paragraph proof of the Inscribed Angle Theorem from the U of Chicago:

Given: Angle ABC inscribed in Circle O
Prove: Angle ABC = 1/2 * Arc AC

Case I: The auxiliary segment OA is required. Since Triangle AOB is isosceles [both OA and OB are radii of the circle -- dw], Angle B = Angle A. Call this measure x. By the Exterior Angle Theorem, Angle AOC = 2x. Because the measure of an arc equals the measure of its central angle, Arc AC = 2x = 2 * Angle B. Solving for Angle B, Angle B = 1/2 * Arc AC. QED Case I.

Notice that the trick here was that between the central angle (whose measure equals that of the arc) and the inscribed angle is an isosceles triangle. We saw the same thing happen in yesterday's proof of the Angle Bisector Theorem -- the angle bisector of a triangle is a side-splitter of a larger triangle, and cutting out the smaller triangle from the larger leaves an isosceles triangle behind.

Let's move onto Case II. Well, the U of Chicago almost gives us a two-column proof here, so why don't we complete it into a full two-column proof. For Case II, O is in the interior of Angle ABC.

Statements                                                     Reasons
1. O interior ABC                                           1. Given
2. Draw ray BO inside ABC                            2. Definition of interior of angle
3. Angle ABC = Angle ABD + Angle DBC       3. Angle Addition Postulate
4. Angle ABC = 1/2 * Arc AD + 1/2 * Arc DC 4. Case I and Substitution
5. Angle ABC = 1/2(Arc AD + Arc DC)           5. Distributive Property
6. Angle ABC = 1/2 * Arc AC                         6. Arc Addition Property and Substitution

The proof of Case III isn't fully given, but it's hinted that we use subtraction rather than addition as we did in Case II. Once again, I bring up the Triangle Area Proof -- the case of the obtuse triangle involved subtracting the areas of two right triangles, whereas in the case where that same angle were acute, we'd be adding the areas of two right triangles.

The text mentions a simple corollary of the Inscribed Angle Theorem:

An angle inscribed in a semicircle is a right angle.

The text motivates the study of inscribed angles by considering camera angles and lenses. According to the text, a normal camera lens has a picture angle of 46 degrees, a wide-camera lens has an angle of 118 degrees, and a telephoto lens has an angle of 18. I briefly mention this on my worksheet. But a full consideration of camera angles doesn't occur until the next section of the text, Lesson 15-4 -- but we're only really doing Lesson 15-3 today.

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