26 Locus: A Set of PointsMathematics Florida StandardsMAFS.912.G-GMD.2.4 ... Identify three-dimensionalobjects generated by rotations of two-dimensionalobjects.

MP1,MP3, MP4, MP6

Objective To draw and describe a locus

Getting Read/!

Where should 5am and

Maria Wilson look for o

new apartment that isequidistant from their jobs?

-. i /office

You studied the

distance between

two points inLesson 5-2. If youneed help, lookback.

MATHEMATICAL

PRACTICES In the Solve It, you described the possible locations based on a certain condition. A

locus is a set of points, all of which meet a stated condition. Loci is the plural of locus.

Lesson

Vocabularylocus

Essential Understanding You can use the description of a locus to sketch ageometric relationship.

Problem 1 Describing a Locus in a Plane

What is a sketch and description for each locus of points in a plane?

o the points I cm from a given point C

1 cm -•. . (7Draw a point C. Sketch several points 1 cm fromC. Keep doing so until you see a pattern. Draw thefigure the pattern suggests.

The locus is a circle with center C and radius 1 cm.

Q the points 1 cm from AB

Have you consideredall possibilities?Make sure that the

endpoints as well as thesegment are included inthe sketch.

1 cm Ij^cmDraw AB. Sketch several points on either side of AB. Alsosketch points 1 cm from point A and point B. Keep doing sountil you see a pattern. Draw the figure the pattern suggests.

The locus is a pair of parallel segments, each 1 cm from AB, and two semicircleswith centers at A and B.

806 Chapter 12 Circles

Got It? 1 ■ Reasoning If the question for part (b) asked for the locus of points in a

plane 1 cm from AB, how would the sketch change?

You can use locus descriptions for geometric terms.

The locus of points in the interior of

an angle that are equidistant from the

sides of the angle is an angle bisector.

a

In a plane, the locus of points that are

equidistant from a segment's endpoints isthe perpendicular bisector of the

segment.

V

Sometimes a locus is described by two conditions. You can draw the locus by firstdrawing the points that satisfy each condition. Then find their intersection.

Problem 2 Drawing a Locus for Two Conditions

What is a sketch of the locus of points in a plane that satisfy these conditions?

• the points equidistant from intersecting lines k and m• the points 5 cm from the point where k and m intersect

Lines k and m intersect. Sketch that satisfies

the given conditions

Make a sketch to satisfy the firstcondition. Then sketch the second

condition. Look for the points in common.

Sketch the points in a plane equidistant from lines k and m.These points form two lines that bisect the vertical angles formedby/candm.

Sketch the points in a plane 5 cm from the point wherek and m intersect. These points form a circle.

Indicate the point or set of points that satisfies both conditions.This set of points is A, B, C and D.

^ Got It? 2. What is a sketch of the locus of points in a plane that satisfy these conditions?• the points equidistant from two points X and Y

• the points 2 cm from the midpoint of XY

c PowerGeometry.com Lesson 12-5 Locus: A Set of Points 807

xu:«r

How can making asketch help?Make a sketch of the

points in a plane andthen visualize what the

figure would look like inthree dimensions.

Describing a Locus in Space

o What is the locus of points in space that are c units from a point D?

The locus is a sphere with center at point D and radius c.

O What is the locus of points in space that are 3 cm from a line €?

The locus is an endless cylinder with radius 3 cm and centerline

^ Got It? 3. What is each locus of points?a. in a plane, the points that are equidistant from two parallel linesb. in space, the points that are equidistant from two parallel planes

&Lesson Check

Do you know HOW?What is a sketch and description for each locus of pointsin a plane?

1. points 4 cm from a point X

2. points 2 in. from UV

3. points 3 mm from LM

4. points 1 in. from a circle with radius 3 in.

MATHEMATICAL

Do you UNDERSTAND? LOT PRACTICES5. Vocabulary How are the words/ocus and/ocah'on

related?

6. Compare and Contrast How are the descriptions ofthe locus of points for each situation alike? How arethey different?

• in a plane, the points equidistant frompoints/and X

• in space, the points equidistant from

points JandK

Practice and Problem-Solving Exercises

Practice Sketch and describe each locus of points in a plan

MATHEMATICAL

PRACTICES

1

e.

7. points equidistant from

the endpoints of PQ

^ See Problem 1.

8. points in the interior of /LABC and

equidistant from the sides of /LABC

9. points equidistant from

two perpendicular lines

10. midpoints of radii of a circlewith radius 2 cm

For Exercises 11-15, sketch the locus of points in a plane that satisfy the

given conditions.

11. equidistant from points M and N and on a circle with center Mand radius = | MN12. 3 cm from GH and 5 cm from G, where GH = 4.5 cm

13. equidistant from the sides of Z.PQR and on a circle with center P and radius PQ

^ See Problem 2.

808 Chapter 12 Circles

^ Apply

14. equidistant from both points

A and B and points C and D

15. equidistant from the sides of

/LJKL and on ©C

Describe each locus of points in space.

16. points 3 cm from a point F

18. points 1 in. from plane M

Describe the locus that each blue figure represents.

^ See Problem 3.

17. points 4 cm from DE

19. points 5 mm from PQ

20. 21. 1 j y\ 1

-/•< ■N -0\ \x

22.M !j

N

23. Open-Ended Give two examples of loci from everyday life, one in a planeand one in space.

24. Writing A classmate says that it is impossible to find a point equidistant from threecollinear points. Is she correct? Explain.

25. Think About a Plan Write a locus description of the points highlighted in blueon the coordinate plane.• How many conditions will be involved?• What is the condition with respect to the origin?• What are the conditions with respect to the x- and y-axes?

—

y\ \ _

0 y>cV>Vr'

Coordinate Geometry Write an equation for the locus of points in a planeequidistant from the two given points.

26. A(0,2) andB(2,0) 27. P(l,3) and Q(5.1) 28. Tt2,-3) and V(6,1)Axis

29. Meteorology An anemometer measures wind speed and wind direction.In an anemometer, there are three cups mounted on an axis. Consider apoint on the edge of one of the cups.

a. Describe the locus that this point traces as the cup spinsin the wind.

b. Suppose the distance of the point from the axis of the anemometeris 2 in. Write an equation for the locus of part (a). Use the axis asthe origin.

C PowerGeometry.com Lesson 12-6 Locus: A Set of Points 809

School

12fl «

30. Landscaping The school board plans to construct afountain in front of the school. What are all the possible

locations for a fountain such that the fountain is 8 ft from

the statue and 16 ft from the flagpole?

Make a drawing of each locus.

31. the path of a car as it turns to the right

32. the path of a doorknob as a door opens

33. the path of a knot in the middle of a jump-rope as it is being used

34. the path of the tip of your nose as you turn your head

35. the path of a fast-pitched softball

36. Reasoning Points A and B are 5 cm apart. Do the following loci in a plane haveany points in common?

the points 3 cm from A

the points 4 cm from AB

Illustrate your answer with a sketch.

Coordinate Geometry Draw each locus on the coordinate plane.

37. all points 3 units from the origin 38. all points 2 units from (—1,3)

39. all points 4 units from the y-axis 40. all points 5 units from x = 2

41. all points equidistant from

y = 3andy = —1

43. all points equidistant from

thejc- andy-axes

42. all points equidistant from

x: = 4 and = 5

44. all points equidistant from

jc = 3 and y = 2

45. a. Draw a segment to represent the base of an isosceles triangle. Locate three

points that could be the vertex of the isosceles triangle.

b. Describe the locus of possible vertices for the isosceles triangle.

c. Writing Explain why points in the locus you described are the only possibilities

for the vertex of the isosceles triangle.

46. Describe the locus of points in a plane 3 cm from the points on a circle with

radius 8 cm.

47. Describe the locus of points in a plane 8 cm from the points on a circle with

radius 3 cm.

48. Sketch the locus of points for the air valve on the tire of a bicycle as the bicycle

moves down a straight path.

810 Chapter 12 Circles

challenge 49. In the diagram, Moesha, Jan, and Leandra are seated at uniformdistances around a circular table. Copy the diagram. Shade the points

on the table that are closer to Moesha than to Jan or Leandra.

Playground Equipment Ihink about the path of a child on each piece

of playground equipment. Draw the path from (a) a top view, (b) afront view, and (c) a side view.

50. a swing

52. a corkscrew slide

54. a firefighters' pole

51. a straight slide

53. a merry-go-round

r

SAT/Aa

c;

Standardized Test Prep

55. What are the coordinates of the center of the circle whose equation is

(;c - 9)2 -I- 0 4)2 = 1?

C® (3,-2) CX) (-3.2) (-9,4)

Short

, Response>Mir

CE^ (9, -4)

56. A plane passes through two adjacent faces of a rectangular prism. The plane is

perpendicular to the base of the prism. Which term is the most specific name for a

figure formed by the cross section of the plane and the prism?

CX" square rectangle CHD parallelogram CX)> kite

57. Margie's cordless telephone can transmit up to 0.5 mi from her home. Carol's

cordless telephone can transmit up to 0.25 mi from her home. Carol and Margielive 0.25 mi from each other. Can Carol's telephone work in a region that Margie'scannot? Sketch and label your diagram.

Mixed Review

Write an equation of the circle with center C and radius r.

58. C(6,-10),r=5 59. C(l, 7), r = 6

Find the surface area of each figure to the nearest tenth.

61. 62. ̂4ft

±

12 in.

4^ See Lesson 12-5.

60. C(-8,-l),r= Via

^ See Lesson 11-2.

12ft

In OO, find the area of sector AOB. Leave your answer in terms of it. See Lesson 10-7.

63. OA = 4, mAB = 90 64. OA = 8, mAB = 72 65. OA = 10, mAB = 36

(jPowerGeometry.com I Lesson 12-6 Locus: A Set of Points 811