Wentworth's Plane and Solid Geometry · Book III, Props. VI–XII · Book 4: Book III: Proportion. Similar Polygons
41. Proportion: composition, division and more
Goal: Your student can prove the remaining theorems of proportion: composition, division, composition and division, equal ratios, products, powers and equimultiples.
You'll need
- The book, open to these pages
- Pencil and paper
What's in the book
Propositions VI–XII (§§ 332–341), with the scholium (§ 341) on what proportions of lines mean.
The lesson, step by step
Warm up
Ask: State and prove "alternation".
The new idea
In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.
Like powers of the terms of a proportion are in proportion.
Equimultiples of two quantities are in the same ratio as the quantities themselves.
Props. VI–VIII are composition (a + b : b = c + d : d), division (a − b : b = c − d : d), and the two together.
Draw the figure
Test each with numbers: 2 : 3 = 8 : 12 gives 5 : 3 = 20 : 12 (composition) and 4 : 9 = 64 : 144 (squares).
Walk through the proof
VI and VII: add 1 to (or subtract 1 from) both sides of a/b = c/d.
VIII: divide the result of VI by the result of VII.
IX: call the common ratio r; then a = br, c = dr and so on; add and divide.
X: multiply the equations. XI: raise both sides to the nth power. XII: multiply top and bottom by m.
Your student proves it back
Your student proves Props. VI, IX and XII aloud and on paper.
Wrap up
Read § 341 together: in alternation all four quantities must be of the same kind. Ask why.