Wentworth's Plane and Solid Geometry · Book III, Exs. 331–348 · Book 4: Book III: Proportion. Similar Polygons
56. Problems of computation (2)
Goal: Your student solves numerical problems about chords, tangents and secants.
You'll need
- The book, open to these pages
- Pencil and paper
- A calculator for square roots (optional)
What's in the book
Pages 182–183: numerical exercises 331–348 on chords, tangents, secants and extreme and mean ratio. This finishes Book III.
The lesson, step by step
Warm up
Ask: If a chord is 3 inches from the centre of a circle of radius 5, how long is it? (8 inches.)
Chords
Ex. 331. The distance from the centre of a circle to a chord 10 feet long is 12 feet. Find the distance from the centre to a chord 24 feet long.
First find the radius.
Tangents and secants
Ex. 336. From the end of a tangent 20 inches long a secant is drawn through the centre of the circle. If the exterior segment of this secant is 8 inches, find the radius of the circle.
More
Your student does three or four more from Ex. 332–345, with a sketch for each.
Review Book III
Ask your student to state the three tests of similar triangles, the Pythagorean theorem, and the two-chord theorem.
Wrap up
Go over the answers together.
Answers to the book's problems
Ex. 331: radius 13 ft; distance 5 ft. Ex. 332: chord 6√3, about 10.39 in; from an end of the chord to the ends of the diameter 6 in and 6√3 (about 10.39) in. Ex. 333: tangents 8 in; chord 9.6 in. Ex. 335: radius 5 in; chords 2√5 (about 4.47) in and 4√5 (about 8.94) in. Ex. 336: 21 in. Ex. 337: 133; shortest chord 2√133, about 23.07 in. Ex. 338: 15 in. Ex. 339: external tangent 10√2, about 14.14 in; internal tangent √104, about 10.20 in. Ex. 340: about 6.18 in and 3.82 in (5√5 − 5 and 15 − 5√5). Ex. 341: acute. Ex. 342: 63 ft. Ex. 344: 24 ft (two chords) and 12√6, about 29.39 ft (two chords). Ex. 345: 4 in (the external segment is 3 in). Ex. 347: (p ± √(8d² − p²))/4.
Worked out for this site from the scan. If a number in your copy differs, trust the book.