Wentworth's Plane and Solid Geometry · Book V, Exs. 557–603 · Book 6: Book V: Regular Polygons and Circles
79. Miscellaneous loci and problems
Goal: Your student finds loci and solves construction problems drawing on all of plane geometry.
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 248–250: problems of loci (Ex. 557–572) and constructions (Ex. 573–603). This is the end of plane geometry.
The lesson, step by step
Warm up
Ask: What is the locus of points at a given distance from a given line?
Loci
Do several quickly with sketches:
Ex. 559. Find the locus of the extremity of a tangent to a given circle, if the length of the tangent is equal to a given line.
Ex. 562. Find the locus of the vertex of a triangle that has a given base and a given altitude.
Constructions
Your student picks two or three of Ex. 573–603.
Ex. 576. To divide a given straight line into two segments such that their product shall be the maximum.
Final review
Ask your student to choose the five most important theorems of plane geometry and state them. Talk about why each matters.
Wrap up
Plane geometry is finished. Celebrate! Solid geometry (Book VI) begins on page 251.
Answers to the book's problems
Ex. 559: a circle concentric with the given one, with radius √(R² + t²), where t is the given length. Ex. 561: a line parallel to the given line, halfway between it and the point. Ex. 562: two lines parallel to the base at the given distance. Ex. 563: if the length exceeds the distance between the parallels, two lines parallel to them; if it equals it, the whole strip between them. Ex. 576: bisect it (the maximum rectangle with a given perimeter is a square).
Worked out for this site from the scan. If a number in your copy differs, trust the book.