Wentworth's Plane and Solid Geometry · Book II, Exs. 149–177 · Book 3: Book II: The Circle
37. Loci and locating points
Goal: Your student finds loci of points and centres, and uses two loci to locate a required point.
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 129–131: Exercises 149–177 on loci: centres of circles meeting given conditions, points equidistant from given points and lines, and moving-figure loci.
The lesson, step by step
Warm up
Ask: What two loci do we know so far? (The perpendicular bisector and the angle bisector.)
Simple loci
Do Ex. 149–155 orally with quick sketches. For example:
Ex. 151. Which has a given radius r and touches a given line AB.
(These ask for the locus of the centre of a circle.)
Two loci locate a point
Many problems are solved by finding two loci and taking their crossing:
Ex. 157. To find a point X equidistant from three given points.
Your student constructs Ex. 157, 160 and 162.
A real-world one
Ex. 167. A straight railway passes two miles from a town. A place is four miles from the town and one mile from the railway. To find by construction the places that answer this description.
Use 1 cm for a mile.
Moving figures
Ex. 170–177 ask for the path traced by a point. Try Ex. 176 with a real ruler sliding with its ends on two edges of a book: mark the middle point as it moves.
Wrap up
Your student explains how two loci together can pin down a point.
Answers to the book's problems
Ex. 149: two circles concentric with the given one, with radii increased and decreased by the distance. Ex. 150: a circle with centre P and radius r. Ex. 151: two lines parallel to AB at distance r. Ex. 152: the perpendicular bisector of PQ. Ex. 153: the perpendicular to AB at P. Ex. 154: the line midway between the parallels. Ex. 155: the two bisectors of the angles formed by the lines. Ex. 157: the meeting point of the perpendicular bisectors (the circum-centre). Ex. 167: up to four places, where the circle of radius 4 about the town meets the two lines parallel to the railway at 1 mile. Ex. 170: given in the book (§ 290). Ex. 176: a quarter circle (arc of a circle) with centre at the corner and radius half the rod.
Worked out for this site from the scan. If a number in your copy differs, trust the book.