hsm11gmse 1206 t07063 - Mrs. Meyer's Math...

Problem 1

806 Chapter 12 Circles

12-6 Locus: A Set of Points

Objective To draw and describe a locus

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.

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

Lesson Vocabulary

•locus

LessonVocabulary

Describing a Locus in a Plane

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

A the points 1 cm from a given point C

Draw a point C. Sketch several points 1 cm from C. Keep doing so until you see a pattern. Draw the figure the pattern suggests.

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

B the points 1 cm from AB

Draw AB. Sketch several points on either side of AB. Also sketch points 1 cm from point A and point B. Keep doing so until 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 semicircles with centers at A and B.

hsm11gmse_1206_t07060

CC1 cm

hsm11gmse_1206_t07061

1 cm 1 cm

A BHave you considered all possibilities?Make sure that the endpoints as well as the segment are included in the sketch.

You studied the distance between two points in Lesson 5-2. If you need help, look back.

Where should Sam and Marla Wilson look for a new apartment that is equidistant from their jobs?

A St.

Marla’soffice

Sam’soffice

B St.

C St.

1st St.

2nd St.

3rd St.

4th St.

Broadway

hsm11gmse_1206_a07944Street map with Sam and Marla’s Office15p x 7p

2nd proof1.13.08

MATHEMATICAL PRACTICES

G-GMD.B.4 . . . Identify three-dimensional objects generated by rotations of two-dimensional objects.

MP 1, MP 3, MP 4, MP 6

Common Core State Standards

Problem 2

Lesson 12-6 Locus:ASetofPoints 807

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 In a plane, the locus of points that are an angle that are equidistant from the equidistant from a segment’s endpoints is sides of the angle is an angle bisector. the perpendicular bisector of the segment.

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

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

Make a sketch to satisfy the first condition. Then sketch the second condition. Look for the points in common.

Sketch that satisfies the given conditions

Lines k and m intersect.

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

Sketch the points in a plane 5 cm from the point where k 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.

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

Got It?

hsm11gmse_1206_t07062

k

m

hsm11gmse_1206_t07064.ai

k

m


k

m


C

D

B

A

Got It?

hsm11gmse_1206_t07063

A

B

Lesson Check

Problem 3


Describing a Locus in Space

A 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.

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

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

3. What is each locus of points? a. in a plane, the points that are equidistant from two parallel lines b. in space, the points that are equidistant from two parallel planes

Got It?

Do you know HOW?What is a sketch and description for each locus of points in 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.

Do you UNDERSTAND? 5. Vocabulary How are the words locus and location

related?

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

• in a plane, the points equidistant from points J and K

• in space, the points equidistant from points J and K

How can making a sketch help?Make a sketch of the points in a plane and then visualize what the figure would look like in three dimensions.

Practice and Problem-Solving Exercises

Sketch and describe each locus of points in a plane.

7. points equidistant from 8. points in the interior of ∠ABC and

the endpoints of PQ equidistant from the sides of ∠ABC

9. points equidistant from 10. midpoints of radii of a circle two perpendicular lines with 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 M and radius = 12 MN

12. 3 cm from GH and 5 cm from G, where GH = 4.5 cm

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

PracticeA See Problem 1.

See Problem 2.




14. equidistant from both points 15. equidistant from the sides of A and B and points C and D ∠JKL and on }C

Describe each locus of points in space.

16. points 3 cm from a point F 17. points 4 cm from <DE

>

18. points 1 in. from plane M 19. points 5 mm from PQ>

Describe the locus that each blue figure represents.

20. 21. 22.

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

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

25. Think About a Plan Write a locus description of the points highlighted in blue on 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?

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

26. A(0, 2) and B(2, 0) 27. P(1, 3) and Q(5, 1) 28. T(2,-3) and V(6, 1)

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

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

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


AO

B C

D


K

J

L

C

See Problem 3.

ApplyB


A


O

y

x1

�1 1�1


a

a

M

N


y

xO1

�1 1�1

AxisSTEM


30. Landscaping The school board plans to construct a fountain 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 have any 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 42. all points equidistant from y = 3 and y = -1 x = 4 and x = 5

43. all points equidistant from 44. all points equidistant from the x- and y-axes x = 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.

hsm11gmse_1206_a07946Landscape map of School and Fountain12p x 6p

3rd proof1.13.08

School

Flagpole20 ft

12 ft

Statue


49. In the diagram, Moesha, Jan, and Leandra are seated at uniform distances 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 Think about the path of a child on each piece of playground equipment. Draw the path from (a) a top view, (b) a front view, and (c) a side view.

50. a swing 51. a straight slide

52. a corkscrew slide 53. a merry-go-round

54. a firefighters’ pole

HSM11GMSE_1206_a079481st pass 11 -24-08Durke

ChallengeC

Mixed Review

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

58. C(6, -10), r = 5 59. C(1, 7), r = 6 60. C(-8, -1), r = 113

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

61. 62.

In }O, find the area of sector AOB. Leave your answer in terms of P.

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

See Lesson 12-5.

See Lesson 11-2.

12 in.

13 in.

15 in.


12 ft

4 ft


See Lesson 10-7.

Standardized Test Prep

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

(x - 9)2 + (y + 4)2 = 1?

(3,-2) (-3, 2) (-9, 4) (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?

square rectangle parallelogram 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 Margie live 0.25 mi from each other. Can Carol’s telephone work in a region that Margie’s cannot? Sketch and label your diagram.

SAT/ACT

ShortResponse

hsm11gmse 1206 t07063 - Mrs. Meyer's Math...

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