Wentworth's Plane and Solid Geometry · Book II, Exs. 208–245 · Book 3: Book II: The Circle
39. Constructions with quadrilaterals and tangent circles
Goal: Your student constructs rectangles, rhombuses and trapezoids from given parts, and circles tangent to given lines and circles.
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 132–134: Exercises 208–245, the last of Book II, on quadrilaterals, circles of given radius touching given figures, and common tangents to two circles.
The lesson, step by step
Warm up
Ask: What is the locus of the centre of a circle of radius r touching a given line?
Quadrilaterals
Your student constructs one or two:
Ex. 212. The two diagonals.
(Ex. 212 is a rhombus; its diagonals bisect each other at right angles.)
Circles of given radius
Ex. 227. Touches each of two intersecting lines AB and CD.
Use two loci for the centre: lines parallel to each given line at distance r.
Common tangents
Read Ex. 245 and its printed method. Draw two circles of radii 3 cm and 1.5 cm, 7 cm apart, and construct the outer common tangents following the steps.
Review
Your student lists, from memory, the 18 problems of construction in Book II (Props. XXI–XXXVIII).
Wrap up
Book II is finished. Ask which construction your student is proudest of.
Answers to the book's problems
Ex. 212: draw the two diagonals bisecting each other at right angles and join their ends. Ex. 227: the centre is where lines parallel to AB and CD at distance r cross (four solutions).
Worked out for this site from the scan. If a number in your copy differs, trust the book.