Unit 10-Aim 3: How can we determine the solution to compound loci problems?
Motivation:
1. How is this problem different from the previous problems that we have solved?
Jan 2008: 37. A city is planning to build a new park. The park must be equidistant from school A at
3, 3 and school B at 3,  5 .
The park also must be exactly 5 miles from the center of town, which is
located at the origin on the coordinate graph. Each unit on the graph represents 1 mile. On the set of axes
below, sketch the compound loci and label with an X all possible locations for the new park.
Answer: The solution involves two loci.
2. Fill in:
Compound Loci
A solution is a point of intersection of loci.
3. The plural of locus is loci or compound loci.
Solve the problem above.
Step I: Locus
The locus of all points that are equidistant from A
and B is the perpendicular bisector of AB .
Step II: Locus
The locus of all points that are 5 miles from a point
O is a circle with center O and radius 5 miles.
Step III: Compound Loci (label intersections with X )
ANSWER:
4. Jan 2010: 37. On the grid below, graph the points that are equidistant from both the x and y axes and
the points that are 5 units from the origin. Label with an X all points that satisfy both conditions.
Step I: Locus
The locus of all points that are equidistant from
both the x and y axes are two angle bisectors.
Step II: Locus
The locus of all points that are 5 units from a point
O is a circle with center O and radius 5 miles.
Step III: Compound Loci (label intersections with X )
ANSWER:
5. Jan 2011: 35. On the set of axes below, graph the locus of points that are four units from the point
2,1 . On the same set of axes, graph the locus of points that are two units from the line
coordinates of all points that satisfy both conditions.
x  4 . State the
Step I: Locus
The locus of all points that are 4 units from the
point 2,1 is a circle with 2,1 as its center and a
radius of 4 units.
Step II: Locus
The locus of all points that are two units from the
line x  4 are two lines on opposite sides of the
line x  4 , each parallel to the line x  4 and two
units from the line x  4 . These lines are x  2
and x  6
Step III: Compound Loci (label intersections with X )
ANSWERS: 2, 5 , 2,  3 and 6,1
6. Like Aug 2010: 33. Car A is parked 7 miles from car B. Sketch the points that are 4 miles from car A
and sketch the points that are 4 miles from car B. Label with an X all points that satisfy both conditions.
PI GG22 87 Lesson #28
AIM: How do we solve problems using compound loci?
Students will be able to:
1. sketch on the same diagram the locus of points satisfying two given conditions and locate the point(s) of
intersection of these loci
2. explain why the point(s) of intersection represent the locus of points satisfying two conditions
3. determine the number of points satisfying both conditions
Writing Exercise:
1. How is a compound loci problem like a buried treasure map?
2. Create a buried treasure map and a list of the compound loci conditions for finding your buried treasure.
PI GG23 90 Lesson #31
AIM: How do we find the points in the coordinate plane which satisfy two different conditions?
Students will be able to:
1. draw, on the same coordinate axes, the locus of points that satisfy the first of two conditions, the second of two
conditions, and the intersection of these loci
2. explain why the points of intersection represent the locus of points satisfying two conditions
3. determine the number of points satisfying both conditions
Writing Exercise:
Maria solved a compound loci problem. Maria said that after she drew her compound loci diagram she got her
answer by counting all the places where the lines and the circles in her diagram intersected. Explain to Maria why
this technique could get her the wrong answer.
Student Name ____________________________
Extra:
HW3 Aug 2009: 36. On the set of axes below, sketch the points that are 5 units from the origin and
sketch the points that are 2 units from the line y  3 . Label with an X all points that satisfy both
conditions.
Step I: Locus
The locus of all points that are 5 units from the
origin is a circle with the origin 0, 0  as its center
and a radius of 5 units.
Step II: Locus
The locus of all points that are two units from the
line y  3 are two lines on opposite sides of the
line y  3 , each parallel to the line y  3 and two
units from the line y  3 . These lines are y  1
and y = 5.
Step III: Compound Loci (label intersections with X )
ANSWER:
On the set of axes below, sketch the points that are 9 units from the origin and sketch the points that are
equidistant from the x and y axes. Label with an X all points that satisfy both conditions.
Dan is sketching a map of the location of his house and his friend Matthew's house on a set of coordinate
axes. Dan locates his house at point D(0,0) and locates Matthews house, which is 6 miles east of Dan's
house, at point M(6,0). On the accompanying set of coordinate axes, graph the locus of points equidistant
from the two houses and also 4 units from Matthews house. Then mark with an X all points that satisfy
both conditions.
On the set of axes below, sketch the points that are 5 units from the point (2, 3) and sketch the points that
are 2 units from the lines x =  1 and x = 3. Label with an X all the points that satisfy both conditions.