Wentworth's Plane and Solid Geometry · Book II, §§ 319–322, Exs. 147–148 · Book 3: Book II: The Circle
36. How to solve a construction problem
Goal: Your student knows the steps of analysis, construction, proof and discussion, and follows them on the book's two worked problems.
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
Sections 319–322 on solving problems by analysis, and the two fully worked examples, Exercises 147 and 148.
The lesson, step by step
Warm up
Ask: Inscribe a circle in a triangle and explain why it works.
The method
Read §§ 319–322. Recall § 27 from the Introduction: analysis, construction, proof, discussion. The key idea:
1. Suppose the construction made, and let the figure represent all parts concerned, both given and required.
First worked problem
Work through Ex. 147 with your student drawing each step:
Ex. 147. Problem. To construct a circle that shall pass through a given point and cut chords of a given length from two parallels.
The centre must be on the line midway between the parallels. That one locus is the key.
Second worked problem
Ex. 148 (a triangle from its perimeter, one angle and an altitude) is harder. Read the analysis together and ask your student to explain why angle ECF is known.
Discussion
Ask: When is Ex. 147 impossible? (When the given point is farther from the middle line than the radius found.)
Wrap up
Your student writes the four parts of a solution as headings in the notebook, to use from now on.