Locus: a Set of Points 12-6 1

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 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 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 . 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. points 5 mm from PQ

B Apply Your Skills Describe the locus that each blue ﬁgure represents. 18–20. See margin. 18. 19.y 20. M 1 a A O ᎐11x ᎐1 a GPS Guided Problem Solving L3 N Enrichment L4 Reteaching L2 21. Open-Ended Give two examples of loci from everyday life, one in a plane and one in space. Check students’ work. Adapted Practice L1 PracticeName Class Date L3

22. Reasoning Rosie says that it is impossible to ﬁnd a point equidistant from Practice 12-6 Tessellations

Describe the symmetries of each tessellation. Copy a portion of the three collinear points. Is she correct? Explain. See margin. tessellation, and draw any centers of rotational symmetry or lines of symmetry.

1. 2. 3. Real-World Connection Coordinate Geometry Write an equation for the locus: In the plane, the points equidistant from the two given points. A ﬁngertip of the skater traces 4. 5. 6. a locus as she twirls. 23. A(0, 2) and B(2, 0) GPS 24. P(1, 3) and Q(5, 1) 25. T(2, -3) and V(6, 1) y ≠ xy≠ 2x – 4 y ≠–x ± 3

Lesson 12-6 Locus: A Set of Points 703 Identify the repeating ﬁgure or ﬁgures that make up each tessellation. 7. 8.

18. the set of all points on 20. the set of all points a units M, # to AB at its midpt. l Use each ﬁgure to create a tessellation on dot paper. the bisector of A from planes M and N Similarly, pts. equidist. 9. 10. 11. 22. Yes; if the collinear pts. from B and C are on 19. the set of all points 2 © Pearson Education, Inc. All rights reserved. plane N, # at the midpt. units from the origin are A, B, and C, then the Determine whether each ﬁgure will tessellate a plane. n 12. rhombus 13. acute triangle 14. regular decagon locus of pts. equidist. of BC. But M N. 15. regular hexagon 16. regular dodecagon 17. regular 15-gon from A and B is a plane 703 26. Coordinate Geometry Complete the following locus y 4. Assess & Reteach GO nline description of the points highlighted in blue at the right: 2 1 Homework Help in the coordinate plane, the points 2 units from the 9 x PowerPoint 9 ᎐22᎐1 1 Visit: PHSchool.com and 1 unit from the -axis. origin; x ᎐1 Lesson Quiz Web Code: aue-1206 ᎐2 Make a drawing of each locus. 27–30. See margin. In Exercises 1–3 draw and 27. the path of a doorknob as a door opens describe each locus in a plane. 28. the path of a knot in the middle of a jump rope as it is being used 1. 2 cm from O with radius 4 cm 29. the path of the tip of your nose as you turn your head 30. the path of a fast-pitched softball

31. Draw a circle with 31. Jack and Julie Wilson take new O 2 cm center downtown jobs in Shrevetown and need a 6 cm and radius 3 mi. 4 cm place to live. Jack says, “Let’s try Connect Jack’s and to move somewhere equidistant Julie's Julie’sofﬁces with a from both of our ofﬁces.” Julie office segment and construct two concentric circles with says,“Let’s try to stay within the # bis. Locations radii 2 cm and 6 cm three miles of downtown.” will lie on the # bis. Jack's Office 2. equidistant from the vertices and on or inside Where on the map should the of rectangle ABCD the circle. Wilsons look for a home? AB 32. yes; 2 points 32. Critical Thinking Points A and B are 5 cm apart. Do the Downtown following loci in a plane have any points in common? 3 cm 4 cm Scale (mi) the points 3 cm from A C D AB5 cm 0123456 point of intersection of the points 4 cm from B the diagonals of ABCD Illustrate with a sketch. 3. 3 cm from line m and 4 cm 33–40. See back of book. from point P, where P is Coordinate Geometry Draw each locus on the coordinate plane. 6 cm from line m 33. all points 3 units from the origin 34. all points 2 units from (-1, 3) 3 cm m 35. all points 4 units from the y-axis 36. all points 5 units from x = 2 3 cm 37. all points equidistant from 38. all points equidistant from 4 cm y = 3 and y =-1 x = 4 and x = 5 P Spinning cup Axis 39. all points equidistant from 40. all points equidistant from the x- and y-axes x = 3 and y = 2 two points where P with radius 4 cm intersects a line Meteorology In an anemometer, there are three cups mounted on an axis. parallel to and 3 cm from Imagine a point on the edge of one of the cups. line m 41. Describe the locus that this point traces as the cup spins in the wind. a circle 4. Draw and describe the locus: In space, the points 10 in. from a 42. Suppose the distance of the point from the axis of the anemometer is 2 in. segment 30 in. long. Write an equation for the locus of part (a). Use the axis as the origin. x2 ± y2 ≠ 4 43. Robert draws a segment to use as the base of an isosceles triangle. AB a. Draw a segment to represent Robert’s base. Locate three points that could 10 in. be the vertex of the isosceles triangle. a–c. See margin p. 705. the lateral surface of a Real-World Connection b. Describe the locus of possible vertices for Robert’s isosceles triangle. cylinder of radius 10 in. and An anemometer measures c. Writing Explain why points in the locus you described are the only two hemispheres of radius 10 wind speed. possibilities for the vertex of Robert’s triangle. in. with centers at the segment’s endpoints 5. Describe the locus of points 704 Chapter 12 Circles satisfying the equation of a – 2 + + 2 = circle (x 2) (y 8) 25. 27. 28. 29. the set of points in a coordinate plane 5 units from the point (2, –8) top view

top view 30. 704 side view side view 44. Describe the locus: The points in space equidistant from the points of a circle. Alternative Assessment See left. Have each student write a locus C Challenge Playground Equipment Think about the path of a child on each piece of problem with a condition that playground equipment. Draw the path from (a) a top view, (b) a front view, and 44. a line through the must be met in a plane and (c) a side view. 45–49. See margin. center of the circle, another with a condition that # to the plane of the 45. a swing 46. a straight slide 47. a corkscrew slide must be met in space, draw the circle loci, and describe the loci in 48. a merry-go-round 49. a ﬁreﬁghters’ pole Jan writing. Then have students Moesha exchange problems with partners, 50. In the diagram, three students are seated at solve the problems, and compare uniform distances around a circular table. their solutions. Copy the diagram. Shade the points on the table that are closer to Moesha than to Jan or Leandra. See margin. Leandra Test Prep

Test Prep Resources For additional practice with a variety of test item formats: Multiple Choice 51. Which graph shows the locus: The points in a plane 1 unit from the • Standardized Test Prep, p. 711 53. [2] no intersection of the lines x + y = 2 and x - y = 4? B • Test-Taking Strategies, p. 706 A.y B. y • Test-Taking Strategies with 1 2 264 Transparencies x −1 O O 264 x M C C.y D. y 2 −2 2 4 x O 264 x 46. front top 52. Which equation describes the locus: The points in the coordinate plane that are 5 units from the y-axis? G F. Δy«=5 G. Δx«=5 H. x + y = 5 J. x2 + y2 = 25

Short Response 53. Margie’s cordless telephone can transmit up to 0.5 mile from her home. side Carol’s cordless telephone can transmit up to 0.25 mile from her home. Carol and Margie live 0.25 mile from each other. Can Carol’s telephone work in a region that Margie’s cannot? Sketch and label a diagram. [2] See left. [1] incorrect answer OR incorrect diagram. 47. top view

MixedMixed ReviewReview side front view view for Lesson 12-5 Write an equation of the circle with center C and radius r. 54–56. See left. GO 48. Help 54. C(6, -10), r = 5 55. C(1, 7), r = 6 56. C(-8, 1), r = !13 side view front view Lesson 11-2 Find the surface area of each ﬁgure to the nearest tenth.

54. (x – 6)2 ± (y ± 10)2 ≠ 25 57.13 in. 58. top view 55. (x – 1)2 ± (y – 7)2 ≠ 36 4 ft 56. (x ± 8)2 ± (y – 1)2 ≠ 13 12 ft 15 in. 175.9 ft2 49. 12 in. 510 in.2 front side Lesson 10-7 In (O, ﬁnd the area of sector AOB. Leave your answer in terms of π. 0 0 0 = AB = = AB = = AB = 59. OA 4, m 90 60. OA 8, m 72 61. OA 10, m 36 top 4π 64π 10π 5 lesson quiz, PHSchool.com, Web Code: aua-1206 Lesson 12-6 Locus: A Set of Points 705 50.

43. a. c. The vertex must be 45. J equidist. from the M endpoints of the base. side view front view Base These points lie only on the # bis. # top view b. bis. of the base, C except the pt. on the base 705