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?

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1 12-6 Locus: Set of Points ommon ore State Standards G-GMD Identify three-dimensional objects generated by rotations of two-dimensional objects. MP 1, MP 3, MP 4, MP 6 Objective To draw and describe a locus 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? St. St. St. 1st St. Sam s office roadway 2nd St. Marla s office 3rd St. 4th St. MTHEMTIL PRTIES In the Solve It, you described the possible locations based on a certain condition. locus is a set of points, all of which meet a stated condition. Loci is the plural of locus. Lesson Vocabulary locus Essential Understanding You can use the description of a locus to sketch a geometric relationship. Problem 1 Describing a Locus in a Plane What is a sketch and description for each locus of points in a plane? the points 1 cm from a given point 1 cm Draw a point. Sketch several points 1 cm from. Keep doing so until you see a pattern. Draw the figure the pattern suggests. The locus is a circle with center and radius 1 cm. Have you considered all possibilities? Make sure that the endpoints as well as the segment are included in the sketch. the points 1 cm from 1 cm 1 cm Draw. Sketch several points on either side of. lso sketch points 1 cm from point and point. 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, and two semicircles with centers at and. 806 hapter 12 ircles

2 Got It? 1. Reasoning If the question for part (b) asked for the locus of points in a plane 1 cm from < >, 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. In a plane, the locus of points that are equidistant from a segment s endpoints is 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. Problem 2 Drawing a Locus for Two onditions 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 first condition. Then sketch the second condition. Look for the points in common. k m k 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. m D k m Indicate the point or set of points that satisfies both conditions. This set of points is,,, 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 Lesson 12-6 Locus: Set of Points 807

3 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. Problem 3 Describing a Locus in Space 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. 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 /. Got It? 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 Lesson heck 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 UNDERSTND? MTHEMTIL PRTIES 5. Vocabulary How are the words locus and location related? 6. ompare and ontrast 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 Practice and Problem-Solving Exercises MTHEMTIL PRTIES Practice Sketch and describe each locus of points in a plane. See Problem points equidistant from 8. points in the interior of and the endpoints of PQ equidistant from the sides of 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. See Problem equidistant from points M and N and on a circle with center M and radius = 1 2 MN 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 808 hapter 12 ircles

4 14. equidistant from both points 15. equidistant from the sides of and and points and D JKL and on } J K O L D 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 > See Problem 3. pply Describe the locus that each blue figure represents y O Open-Ended Give two examples of loci from everyday life, one in a plane and one in space. x M a a N 24. Writing classmate says that it is impossible to find a point equidistant from three collinear points. Is she correct? Explain. 25. Think bout 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? oordinate Geometry Write an equation for the locus of points in a plane equidistant from the two given points. y 1 O x 26. (0, 2) and (2, 0) 27. P(1, 3) and Q(5, 1) 28. T(2,-3) and V(6, 1) xis STEM 29. Meteorology n anemometer measures wind speed and wind direction. In an anemometer, there are three cups mounted on an axis. onsider 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. Lesson 12-6 Locus: Set of Points 809

5 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? School 12 ft Make a drawing of each locus. Statue 20 ft Flagpole 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 and are 5 cm apart. Do the following loci in a plane have any points in common? the points 3 cm from the points 4 cm from Illustrate your answer with a sketch. oordinate 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 = all points equidistant from 42. all points equidistant from y = 3 and y = -1 x = 4 and x = all points equidistant from 44. all points equidistant from the x- and y-axes x = 3 and y = 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 hapter 12 ircles

6 hallenge 49. In the diagram, Moesha, Jan, and Leandra are seated at uniform distances around a circular table. opy 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 Standardized Test Prep ST/T 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. 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 Short Response 57. Margie s cordless telephone can transmit up to 0.5 mi from her home. arol s cordless telephone can transmit up to 0.25 mi from her home. arol and Margie live 0.25 mi from each other. an arol s telephone work in a region that Margie s cannot? Sketch and label your diagram. Mixed Review Write an equation of the circle with center and radius r. See Lesson (6, -10), r = (1, 7), r = (-8, -1), r = 113 Find the surface area of each figure to the nearest tenth in in. 12 in. 4 ft 12 ft See Lesson In }O, find the area of sector O. Leave your answer in terms of P. See Lesson O = 4, m = O = 8, m = O = 10, m = 36 Lesson 12-6 Locus: Set of Points 811