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A locus is a path of all the points that satisfy a certain condition

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.

NAME  PERIOD  PACKET 8 LOCI A LOCUS


P

NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS 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.

NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS

d

NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS

d

NAME  PERIOD  PACKET 8 LOCI A LOCUS



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.

NAME  PERIOD  PACKET 8 LOCI A LOCUS


NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS

NAME  PERIOD  PACKET 8 LOCI A LOCUS 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.


NAME  PERIOD  PACKET 8 LOCI A LOCUS

NAME  PERIOD  PACKET 8 LOCI A LOCUS


NAME  PERIOD  PACKET 8 LOCI A LOCUS


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.




NAME  PERIOD  PACKET 8 LOCI A LOCUS


NAME  PERIOD  PACKET 8 LOCI A LOCUS

NAME  PERIOD  PACKET 8 LOCI A LOCUS


NAME  PERIOD  PACKET 8 LOCI A LOCUS









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 (2) one line

(3) two parallel lines (4) two intersecting lines


14. The locus of points equidistant from two fixed points is (1) one circle (2) one line

(3) two circles (4) two lines


15. The locus of points equidistant from the four vertices of a given rectangle is (1) one line

(2) two lines (3) one point (4) a circle




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 ____________________

NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS












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)

NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS


























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


NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS
















B. The locus of points equidistant from one line:


___________________________________________________________________________


Example: Give the locus of points 3 units from the line y = 5

NAME  PERIOD  PACKET 8 LOCI A LOCUS















(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) Draw the locus of points 3 units from the y-axis.

(b) Write an equation for the locus of points r units from the point (1, 3).

(c) How many points satisfy both conditions if r = 1?

(d) How many points satisfy both conditions if r = 4?

NAME  PERIOD  PACKET 8 LOCI A LOCUS




















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.


NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS NAME  PERIOD  PACKET 8 LOCI A LOCUS 18 feet NAME  PERIOD  PACKET 8 LOCI A LOCUS


8 feet


NAME  PERIOD  PACKET 8 LOCI A LOCUS



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