Name: _____________________________________________ Period: _______
Packet 8: Loci
A locus is a path of all the points that satisfy a certain condition. You need to be familiar with
these 5 basic loci.
The locus of points equidistant from:
1. A point:
The locus of points a fixed distance, d, from point P is a circle with the given point P as its
center and d as its radius.
P
d
2. A line:
The locus of points a fixed distance, d, from a line is a pair of parallel lines d distance on
either side of the line.
d
d
3. Two points:
The locus of points equidistant from 2 points, P and Q, is the perpendicular bisector of
segment formed by the 2 points.
P
Q
4. Two parallel lines:
The locus of points equidistant from 2 parallel lines is a line parallel to both given lines and
midway between them.
5. Two intersecting lines:
The locus of points equidistant from 2 intersecting lines is a pair of perpendicular lines that
bisect the angles formed by the intersecting lines.
To discover a probable locus:
1. Make a diagram that contains the fixed lines or points that are given.
2. Decide what condition must be satisfied, and locate one point that meets the
given condition.
3. Then locate several other points that satisfy the given condition. These points
should be sufficiently close together to develop the shape or the nature of the
locus.
4. Use the points to draw a line or a smooth curve that appears to be the locus.
5. Describe in words the geometric figure that appears to be the locus.
Practice:
1. What is the locus of the outer end of the hour hand of a clock during a 12-hour period?
2. What is the locus of the center of a train wheel that is moving along a straight, level
track?
3. What is the locus of a car that is being driven down a street equidistant from the two
opposite parallel curbs?
4. A dog is tied to a stake by a rope 6 meters long. Discover the boundary of the surface
over which he may move.
5. A boy walks through an open field that is bounded on 2 sides by straight, intersecting
roads. He walks so that he is always equidistant from the 2 intersecting roads. Describe
his path.
6. There are 2 stationary floats on a lake. A girl swims so that she is always equidistant
from floats. Determine her path.
7. A dime is rolled along a horizontal line so that the dime always touches the line. What
is the locus of the center of the dime?
8. What is the locus of points that is 10 centimeters from a given point?
9. What is the locus of points equidistant from 2 points, A and B, that are 8 meters
apart?
10. What is the locus of points equidistant from 2 parallel lines 6 feet apart?
11. What is the locus of points 4 inches away from line AB?
12. What is the locus of points equidistant from 2 parallel lines that are 6 inches apart?
Select the choice that best completes the statement.
13. The locus of points a given distance from a line is (1) a circle
(3) two parallel lines
(4) two intersecting lines
14. The locus of points equidistant from two fixed points is
one line
(3) two circles
(4) two lines
(2) one line
(1) one circle
15. The locus of points equidistant from the four vertices of a given rectangle is
one line
(2) two lines
(3) one point
(4) a circle
(2)
(1)
Equation of a Circle
Circle:______________________________________________________________
Equation of a circle whose center is the origin: ______________________
Equation of a circle whose center is a fixed point (a, b): _______________________
Examples:
Give the equation of each of the following, then sketch each on graph paper.
1. the locus of points 6 units from the point (2,3) ________________________
2. the locus of points 4 units from (-4,6)
3. the locus of points 9 units from the origin
________________________
____________________
2 Fixed Points: The Perpendicular Bisector
The locus of points equidistant from 2 points, P and Q, is the perpendicular bisector of the
segment determined by the 2 points.
To find the equation of the locus of points equidistant from 2 points:
Plot each pair of points on the grid.
Draw the segment connecting the 2 points.
Locate the midpoint of the segment.
Through the midpoint, draw a line perpendicular to the segment. Give the equation
of that line.
Examples:
Give the locus of points equidistant from each of the following pairs of points.
1. (2, 6) & (2, -2)
2. (3, -4) & (-1, -4)
Locus of Points from Fixed Lines
A. The locus of points equidistant from 2 parallel lines :
___________________________________________________________________________
Examples:
Give the equation of the locus of points equidistant from each pair of parallel lines.
1. x = 2, x = 8
2. y = -6, y = 3
B. The locus of points equidistant from one line:
___________________________________________________________________________
Example: Give the locus of points 3 units from the line y = 5
(Intersection of loci)
A compound locus problem involves two or more locus conditions occurring at the same time.
The different conditions in a compound locus problem are generally separated by the word
“AND” or the words “AND ALSO”.
Steps for solving:
1. Draw a diagram showing the given information in the problem.
2. Read the problem carefully, looking for the words “AND” or “AND ALSO”, and separate the
problem into the 2 locus conditions.
3. Plot the first locus condition. Draw the locus with a dotted line.
4. On the same diagram, plot the 2nd locus condition, drawing it with a dotted line.
5. Where the dotted lines intersect will be the points which satisfy both conditions. These
points of intersection will be the answer to the compound locus problem.
Examples:
1. How many points are equidistant from 2 intersecting lines and also 5 units from their point of
intersection?
Separate the 2 conditions:
How many points are {equidistant from 2 intersecting lines} and also {5 units from their
point of intersection}?
Draw:
2. What is the number of points equidistant from 2 parallel lines and also equidistant from 2
points on one of the lines?
3. (a)
(b)
(c)
(d)
Draw the locus of points 3 units from the y-axis.
Write an equation for the locus of points r units from the point (1, 3).
How many points satisfy both conditions if r = 1?
How many points satisfy both conditions if r = 4?
4. A treasure is buried in your backyard. The picture below shows your backyard which
contains a tree stump, a teepee, and a tree. The teepee is 8 feet from the stump and 18 feet
from the tree (at a right angle). The treasure is equidistant from the teepee and the tree and
also 10 feet from the stump. Locate all possible points of the buried treasure.
18 feet
8 feet