Remember There Are 5 Loci

This assignment is due on Monday May 4th. All questions should be completed on loose leaf accompanied by the diagrams. The last few questions must be on graph paper. We completed every locus except the last one so read about it in the notes packet. I have included its diagram below.

Remember there are 5 loci: 1. from a point: draw a circle 2. from 2 points: draw a line in the middle 3. from a line: draw 2 parallel lines (one on each side) 4. from 2 parallel lines: draw a parallel line in the middle 5. from 2 intersecting lines: draw another pair of intersecting lines

(each of the dotted lines is in the middle so they are angle bisectors)

Look at what is written after the word FROM. It will let you know which loci you are working with. Sometimes the problem doesn’t specify point or line. Just note that trees, statues, poles, fire hydrants, etc are considered points and that a fence is a line.

If you have any questions, please email me and I will try to guide you in your diagram formations. You will be quizzed and tested on locus next week. NAME______DATE______GEOMETRY LOCUS MRS. BINASO

Draw a diagram for each of the following on graph paper or loose leaf. (Assignment will be unacceptable if not accompanied with diagrams.) PART I: 1. The locus of the midpoints of the radii of a circle is (1) a point (2) two lines (3) a line (4) a circle

2. How many points are equidistant from two intersecting lines, l and m, and 3 units from the point of intersection of the lines? (1) 1 (2) 2 (3) 3 (4) 4

3. The distance between points P and Q is six units. How many points are equidistant from P and Q and also three units from P? (1) 1 (2) 2 (3) 3 (4) 6

4. The distance between points P and Q is 8 units. How many points are equidistant from P and Q and also 3 units from P? (1) 1 (2) 2 (3) 0 (4) 4

5. The distance between parallel lines l and m is 12 units. Point A is on line l. How many points are equidistant from lines l and m and 8 units from point A? (1) 1 (2) 2 (3) 3 (4) 4

6. In the coordinate plane, what is the total number of points 5 units from the origin and equidistant from both the x- and y-axes? (1) 1 (2) 2 (3) 0 (4) 4

7. How many points are equidistant from two intersecting lines and 3 units from their point of intersection? (1) 1 (2) 2 (3) 3 (4) 4

8. An equation of the locus of points equidistant from the points (0,6) and (0,-2) is (1) x = 2 (2) y = 2 (3) x = -2 (4) y = -2

9. If the graphs of x2 + y2 = 4 and y = -4 are drawn on the same set of axes, what is the total number of points common to both graphs? (1) 1 (2) 2 (3) 3 (4) 0

10. Which is an equation of the locus of points that are 3 units from the point (3,2)? (1) (x  3)2 + (y  2)2 = 9 (3) (x  3)2 + (y  2)2 = 3 (2) (y  2) = (x  3) + 3 (4) y = 3x  7

11. What is the total number of points that the graphs of x2 + y2 = 16 and y = x have in common? (1) 1 (2) 2 (3) 0 (4) 4 12. Points M and N lie on line l. Line k is parallel to line l. The total number of points equidistant from points M and N and also equidistant from lines l and k is (1) 1 (2) 2 (3) 0 (4) 4

13. Which equation represents the locus of points equidistant from the points (1,1) and (7,1)? (1) x = 4 (2) y = 4 (3) x = -4 (4) y = -4

14. If the graphs of the equations x2 + y2 = 25 and y = x are drawn on the same set of axes, what is the total number of points common to both graphs? (1) 1 (2) 2 (3) 3 (4) 0

15. What is the locus of points at a given distance from a given line? (1) 1 point (2) 2 points (3) 1 circle (4) 2 parallel lines

16. The equation of the locus of points 5 units from the origin is: (1) x2 + y2 = 5 (2) x2 + y2 = 25 (3) x = 5 (4) y = 5

17. An intersection of the graphs of the equations y = -x and y = x2  2 is (1) (-1,-1) (2) (-2,2) (3) (2,2) (4) (1,1)

18. Which is an equation of the locus of points equidistant from the points (-2,0) and (4,0)? (1) x = 1 (2) x = -1 (3) y = 1 (4) y = -1

19. When drawn on the same set of axes, which pair of equations will result in two points of intersection? (1) y = x and y = -x (2) y = 3 and x = -3 (3) y = x2 and y =-x2 (4)y = x and x2 + y2 = 1

20. Which equation represents the locus of points equidistant from the line whose equations are y = 3x + 8 and y = 3x  6? (1) y = 3x  1 (2) y = 3x + 1 (3) y = 3x  4 (4) y = 3x + 4

21. Which equation represents the locus of points equidistant from the points (4,2) and (8,2)? (1) x = 6 (2) y = 6 (3) x = 12 (4) y = 12

22. If point P is on line l, what is the total number of points 3 centimeters from point P and 4 cm from line l? (1) 1 (2) 2 (3) 0 (4) 4

23. If the graphs of the equations x2 + y2 = 16 and y = 4 are drawn on the same set of axes, what is the total number of points common to both graphs? (1) 1 (2) 2 (3) 3 (4) 0

24. Lines l and m are parallel lines 8 cm apart and point P is on line l. What is the total number of points that are equidistant from lines l and m and 5 cm from P? (1) 1 (2) 2 (3) 0 (4) 4 PART II: 25. Maria's backyard has two trees that are 40 feet apart, as shown in the accompanying diagram. She wants to place lampposts so that the posts are 30 feet from both of the trees. How many locations for the lampposts are possible?

26. Steve has a treasure map, represented in the accompanying diagram, that shows two trees 8 feet apart and a straight fence connecting them. The map states that the treasure is buried 3 feet from the fence and equidistant from the two trees. a. Using a separate sheet of paper, show all the places where the treasure could be buried. Clearly indicate where the treasure could be buried. b. What is the distance between the treasure and one of the trees?

27. A treasure map shows a treasure hidden in a park near a tree and a statue. The map indicates that the tree and the statue are 10 feet apart. The treasure is buried 7 feet from the base of the tree and also 5 feet from the base of the statue. How many places are possible locations for the treasure to be buried? Draw a diagram of the treasure map, and indicate with an X each possible location of the treasure.

28. An architect is deciding where to install a fountain in a long rectangular courtyard. The fountain should be equidistant from the north and south sides of the courtyard. It should also be 30 feet away from the entrance on the north side. The courtyard is 60 feet across. Create a map showing where the fountain should be installed.

Map of the Courtyard North Entrance

60 ft.

South Entrance

29. a) On graph paper, draw the intersecting lines whose equations are y = 2x and y = -2x + 4. b) Draw the locus of points equidistant from these two intersecting lines. c) Write an equation of the locus described in part b.

30. a. On graph paper, draw the intersecting lines whose equations are y = 1/2 x +1 and y = -1/2 x + 5. b) Draw the locus of points equidistant from these two intersecting lines. c) Write an equation of the locus described in part b.

31. Write an equation of the locus of points equidistant from (0,-3) and (0,7). 32. a. Draw the locus of points equidistant from the points (4,1) and (4,5) and write an equation for this locus. b. Draw the locus of points equidistant from the points (3,2) and (9,2) and write an equation for this locus. c. Find the number of points that satisfy both conditions stated in a and b. Give the coordinates of each point found.

33. a. Represent graphically the locus of points: (1) 3 units from the line x = 1 (2) 4 units from the line y = -2 b. Write the equations for the loci represented in a. c. Find the coordinates of the points of intersection of these loci.

34. a. Represent graphically the locus of points: (1) 8 units from the y-axis (2) 10 units from the origin b. Write equations for the loci represented in a. c. Find the coordinates of the points of intersection of these loci.

35. a. Draw the locus of points equidistant from the circles whose equation are x2 + y2 = 4 and x2 + y2 = 36. Write an equation of the locus. b. Draw the locus of points 4 units from the x –axis. Write an equation of the locus. c. Find the number of points that satisfy both conditions stated in a and b. Write the coordinates of each of the points found.