Wentworth's Plane and Solid Geometry · Book III, §§ 323–331 (Props. I–V) · Book 4: Book III: Proportion. Similar Polygons
40. Proportion: first theorems
Goal: Your student knows the vocabulary of proportion and can prove the first five theorems on proportions with algebra.
You'll need
- The book, open to these pages
- Pencil and paper
What's in the book
The definitions of proportion, terms, means and extremes, fourth, third and mean proportionals (§§ 323–326), and Propositions I–V (§§ 327–331).
The lesson, step by step
Warm up
Ask: What is a ratio? Write the ratio 6 : 9 as a fraction in lowest terms.
The new idea
Read §§ 323–326. In a : b = c : d, a and d are the extremes, b and c the means.
In every proportion the product of the extremes is equal to the product of the means.
The mean proportional between two quantities is equal to the square root of their product.
Props. III, IV and V: from ad = bc we may write a proportion; and a proportion can be rearranged by alternation (a : c = b : d) and by inversion (b : a = d : c).
Draw the figure
No figure today. Write a real proportion to test each theorem: 2 : 3 = 8 : 12. Check that 2 × 12 = 3 × 8, that 2 : 8 = 3 : 12, and that 3 : 2 = 12 : 8.
Walk through the proof
I: write the proportion as two equal fractions and clear of fractions.
II: from a : b = b : c, the product of the extremes is b², so b = √(ac).
III: divide both sides of ad = bc by bd.
IV (alternation): multiply both sides of a/b = c/d by b/c.
V (inversion): divide both sides of bc = ad by ac.
Your student proves it back
Your student proves each of the five theorems on the board without the book, in algebra, giving the reason for each step.
Wrap up
Ask: What is the mean proportional between 4 and 9? (6.)