Wentworth's Plane and Solid Geometry · Book III, Exs. 290–311 · Book 4: Book III: Proportion. Similar Polygons
54. Construction problems with proportions
Goal: Your student solves construction problems using proportional lines and similar figures.
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)
- A protractor
What's in the book
Pages 178–179: construction exercises 290–311, including inscribing a square in a triangle and in a semicircle.
The lesson, step by step
Warm up
Ask: Construct the mean proportional between two lines.
A short one
Ex. 290. To divide one side of a given triangle into segments proportional to the adjacent sides (§ 348).
A classic
Ex. 303. To inscribe a square in a given triangle.
Follow the book's hint: build a square outward on the base first, then use similar triangles to shrink it.
Your student's choice
Your student picks one more: Ex. 302, 305 or 309.
Ex. 302. To inscribe a square in a semicircle.
Check by measuring
After each construction, measure to check it.
Wrap up
Your student explains the analysis for Ex. 303 in words.
Answers to the book's problems
Ex. 290: bisect the opposite angle (§ 348). Ex. 302: draw a square on the diameter outward, join the centre to its far corners; where these lines cut the arc are two corners of the required square. Ex. 309: divide the given sum in the given ratio (Prop. XXXVI).
Worked out for this site from the scan. If a number in your copy differs, trust the book.