Wentworth's Plane and Solid Geometry · Book V, Exs. 531–556 · Book 6: Book V: Regular Polygons and Circles
78. Miscellaneous theorems
Goal: Your student proves a selection of theorems from across Books I–V.
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 246–247: miscellaneous theorem exercises 531–556, drawing on the whole of plane geometry.
The lesson, step by step
Warm up
Ask your student to name one theorem from each Book.
Together
Ex. 533. The two lines that join the middle points of the opposite sides of a quadrilateral bisect each other.
Hint: use Ex. 74 (the midpoints form a parallelogram) and § 184.
On their own
Your student picks two or three: Ex. 531, 536, 538, 547 or 553.
Ex. 547. The area of a triangle is equal to half the product of its perimeter by the radius of the inscribed circle.
Discuss
For each, ask: Which Book did the key idea come from?
Wrap up
Your student reads one proof aloud.
Answers to the book's problems
Ex. 533: the midpoints form a parallelogram (Ex. 74), whose diagonals bisect each other (§ 184). Ex. 538: the perpendicular from the centre bisects both chords (§ 245); subtract. Ex. 547: join the in-centre to the vertices; three triangles each with altitude r. Ex. 553: the dodecagon is 12 triangles, each with two sides R and an included angle of 30°, so each has area ¼R².
Worked out for this site from the scan. If a number in your copy differs, trust the book.