Wentworth's Plane and Solid Geometry · Book II, Props. XV–XVI · Book 3: Book II: The Circle
28. Finding a ratio, and angles measured by arcs
Goal: Your student can find the ratio of two lines by repeated laying-off, and prove that central angles are in the ratio of their arcs (both cases).
You'll need
- The book, open to these pages
- A ruler (straightedge) and a pair of compasses
- Sharp pencil, eraser and plain paper (or a geometry notebook)
What's in the book
Proposition XV (§ 286), a problem, and Proposition XVI (§ 287), the first proof using limits, with § 288 explaining what "measured by its arc" means.
The lesson, step by step
Warm up
Ask: What is the test for a limit? State § 284.
The new idea
To find the ratio of two straight lines.
In the same circle or in equal circles, two central angles have the same ratio as their intercepted arcs.
Draw the figure
For Prop. XV: draw two lines, AB about 13 cm and CD about 5 cm. Use the compasses to lay CD off on AB, then the remainder on CD, and so on, as the book does.
For Prop. XVI: draw two equal circles with central angles ACB and A'C'B'.
Walk through the proof
XV: this is the same process as finding a greatest common divisor. In the book's example the last remainder KD fits exactly, and AB : CD = 41 : 13.
XVI, Case 1 (commensurable): a common measure fits 7 times in one arc and 4 in the other. Radii to the division points split the angles into 7 and 4 equal parts, so the angles have the ratio 7 : 4 too.
Case 2 (incommensurable): cut off a commensurable part A'D. The ratios are equal for A'D; as the parts get smaller, the leftover DB' approaches zero, and the equal ratios approach equal limits (§ 284).
Your student proves it back
Close the book. Your student redraws the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("because the radii are equal", "by § 143"). Point to each line or angle as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Case 2 is hard. It is enough at first if your student can explain the idea: make the leftover smaller than anything you like.
Wrap up
Read § 288 together: why we may say "a central angle is measured by its intercepted arc".