12-6 Locus: A Set of Points 12-6 1. Plan Objectives What You’ll Learn Check Skills You’ll Need GO for Help Lessons 1-7 and 3-8 1 To draw and describe a locus • To draw and describe a Sketch each of the following. 1–3. See back of book. locus Examples 1. the perpendicular bisector of CD . And Why ) 1 Describing a Locus in a Plane 2. &EFG bisected by FH 2 Drawing a Locus for Two To interpret a locus Conditions description of a geometric 3. line k parallel to line m and perpendicular to line w, all in plane N figure, as in Example 3 3 Describing a Locus in Space New Vocabulary • locus Math Background 1 Drawing and Describing a Locus A locus is the set of all the points, 1 and only those points, that satisfy a stated condition. The word locus A locus is a set of points, all of which meet a stated condition. To sketch a locus, signals that the characterization draw points of the locus until you see a pattern. of a set will follow. 1 EXAMPLE Describing a Locus in a Plane More Math Background: p. 660D a. Draw and describe the locus: In a plane, the points 1 cm from a given point C. Lesson Planning and Draw a point C. Resources Sketch several points 1 cm from C. 1 cm C C Keep doing so until you see a pattern. See p. 660E for a list of the Draw the figure the pattern suggests. resources that support this lesson. The locus is a circle with center C and radius 1 cm. PowerPoint b. Draw and describe the locus: In a plane, AB Bell Ringer Practice Real-World Connection the points 1 cm from a segment . 1 cm 1 cm The locus is Check Skills You’ll Need The locus of footprints AB of children pushing the • two segments parallel to and AB For intervention, direct students to: merry-go-round is a circle. • two semicircles centered at A and B Constructing Parallel and * ) Perpendicular Lines Quick Check 1 Draw and describe the locus: In a plane, the points 2 cm from a line XY. Lesson 3-8: Examples 1, 4 See left. Extra Skills, Word Problems, Proof 4 Practice, Ch. 3 1.Two lines n to XY, each You can use locus descriptions for geometric terms. 4 Constructing the Angle Bisector 2 cm from XY. A Lesson 1-7: Example 5 2 cm Extra Skills, Word Problems, Proof Practice, Ch. 1 XY2 cm B An angle bisector: The points in A perpendicular bisector of a segment: the interior of the angle that are In a plane, the points that are equidistant from the sides of the angle. equidistant from the segment endpoints. Lesson 12-6 Locus: A Set of Points 701 Special Needs L1 Below Level L2 Before Example 3, have students cut out a large circle Students should have compasses and rulers available and tape a straw along its diameter. Have students as they work through Examples 1 and 2. rotate the straw so they can see that the circle traces out a sphere in three dimensions. learning style: tactile learning style: tactile 701 Sometimes a locus is described by two conditions. You can draw the locus by first Vocabulary Tip 2. Teach drawing the points that satisfy each condition. Then find their intersection. The word locus comes from the Latin word for “location.” Its plural is 2 EXAMPLE Drawing a Locus for Two Conditions Guided Instruction loci (LOH sy). Draw the locus: In a plane, the points equidistant from two lines k and m and 5 cm from the point where k and m intersect. 3 EXAMPLE Math Tip Have students compare these The points in a plane equidistant from k space loci with the analogous lines k and m are two lines that bisect plane loci in Example 1. the vertical angles formed by k and m. m 2. PowerPoint k 2 cm The points in a plane 5 cm from the point Additional Examples where k and m intersect is a circle. m X 2 cm Y For Exercises 1–3, check students’ drawings. B The locus that satisfies both conditions k C is the set of points A, B, C, and D. A 1 Draw and describe the locus: m In a plane, the points 3 cm from D ᭪C with radius 3 cm. center C Quick Check and a concentric circle with 2 Draw the locus: In a plane, the points equidistant from two points X and Y radius 6 cm and 2 cm from the midpoint of XY. See left. 2 Point P is 10 in. from point Q. Draw and describe the locus: In a A locus in a plane and a locus in space can be quite different. plane, the points 6 in. from point P and 8 in. from point Q. EXAMPLE Describing a Locus in Space intersection of ᭪P with radius 3 ᭪ 6 in. and Q with radius 8 in. a. Draw and describe the locus: In space, the points that are c units from a point D. 3 Describe the locus: In space, c the points 4 cm from a plane M. two distinct planes, each 4 cm D from and parallel to plane M The locus is a sphere with center at point D and radius c. Resources • Daily Notetaking Guide 12-6 b. Draw and describe the locus: In space, the points that are 3 cm from a line O. L3 Real-World Connection 3 cm • Daily Notetaking Guide 12-6— A soap bubble is a sphere, a Adapted Instruction L1 locus of points in space that 3 cm ᐉ are a given distance from a given point. The locus is an endless cylinder with radius 3 cm and center-line O. Closure Quick Check 3 Draw and describe each locus. a–b. See back of book. a. In a plane, the points that are equidistant from two parallel lines. Point A lies on line ᐉ in plane P. b. In space, the points that are equidistant from two parallel planes. Draw and describe the locus: In a plane, the points 5 cm from point A You can also think of a locus as a path. For example, the locus of the tip of a hand • on line ᐉ. of a clock each day is the circle traced by the tip as it travels around the clock face. • in plane P. The locus of a point on the handle of a sliding-glass door when you enter a room is • in space. the line segment along which the point travels as the door slides back and forth. Check students’ drawings; two collinear points 5 cm each side of A; circle of radius 5 cm with center A; sphere of radius 5 cm 702 Chapter 12 Circles with center A. Advanced Learners L4 English Language Learners ELL After students complete Example 3, have them draw Point out that the term locus comes from the Latin and describe the set of points in space that are word for location and that its plural is loci, which is equidistant from each of the points on ᭪O. pronounced low-sigh. 702 learning style: verbal learning style: verbal EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. 3. Practice Practice and Problem Solving Assignment Guide A Practice by Example Draw and describe each locus in a plane. 1–8. See back of book. 1 AB1-44 1. points 4 cm from a point X 2. points 2 in. from a segment UV Example 1 C Challenge 45-50 (page 701) * ) for 3. points 3 mm from a line LM 4. points 1 in. from a circle with radius 3 in. GO Test Prep 51-53 Help 5. points equidistant from 6. points in the interior of &ABC and Mixed Review 54-61 the endpoints of PQ equidistant from the sides of &ABC 7. points equidistant from 8. midpoints of radii of a circle Homework Quick Check two perpendicular lines with radius 2 cm To check students’ understanding of key skills and concepts, go over Example 2 In a plane, draw the locus whose points satisfy the given conditions. Exercises 10, 16, 24, 31, 42. (page 702) 9. equidistant from points M and N and 1 Exercises 1–13 Suggest that on a circle with center M and radius = 2 MN See left. 9. the single pt. L students use compass and 10. 3 cm from GH and 5 cm from G, where GH = 4.5 cm straightedge when appropriate. 10–13. See back of book. 11. equidistant from the sides of &PQR and Teaching Tip on a circle with center P and radius PQ Exercises 14–17 Have students MLN 12. equidistant from both points 13. equidistant from the sides of compare each locus with the locus A and B and points C and D &JKL and on ᭪C in a plane that meets the stated B conditions. C J Error Prevention! K C A Exercise 17 Students who correctly O L describe the cylinder may overlook D the points 5 mm from the endpoint P. Remind them to consider point P separately to help Draw and describe each locus in space. 14–17. See back of book. Example 3 * ) them see that part of the locus is (page 702) DE 14. points 3 cm from a point F 15. points 4 cm from a line) a hemisphere. 16. points 1 in. from plane M 17.