Name: ____________________________________ Date: __________________

LOCUS Day 1 – Basic Loci A LOCUS is a ___________________________________________________ which satisfies a certain condition. As seen by the headlights (and taillights) in the picture on the left, a locus of points (the headlights or taillights) is the path traced by the moving points under given conditions (following the road).

Think of a locus as a "bunch" of points that all do the same thing. In Latin, the word locus means place. The plural of locus is loci.

There are some basic locus theorems (rules). Each theorem will be explained in detail in the following sections under this topic. Even though the theorems sound confusing, the concepts are easy to understand. We are going to explore these theorems now.

Locus Theorem Number 1: The locus of points at a fixed distance, d, from point P is

_____________________________________________________________________________________

_____________________________________________________________________________________

Picture:

Locus Theorem Number 2: The locus of points at a fixed distance, d, from a line l is

_____________________________________________________________________________________

_____________________________________________________________________________________

Picture:

P

l

Locus Theorem Number 3: The locus of points equidistant from two points, P and Q, is

_____________________________________________________________________________________

_____________________________________________________________________________________

Picture:

Locus Theorem Number 4: The locus of points equidistant from two parallel lines, l1

and l2, is ___________________________________________________________________________

____________________________________________________________________________________

Picture:

Locus Theorem Number 5: The locus of points equidistant from two intersecting lines,

l1 and l2, is _________________________________________________________________________

____________________________________________________________________________________

Picture:

P Q

l1

l2

l1

l2

Locus Theorem Number 6: The locus of all points d units from a circle having a

center at P and a radius of r is _______________________________________________

______________________________________________________________________________

Locus Theorem Number 7: The locus of all points equidistant from two

concentric circles having a center at P and radii of p and q units is ___________

______________________________________________________________________________

Locus Theorem Number 8: The locus of all points equidistant from the sides of

a given angle is _____________________________________________________________

P r

Steps to solving a locus problem:

1. Draw a diagram showing the given lines and points.

2. Read carefully to determine the needed condition(s).

3. Locate one point that satisfies the needed condition and plot it on your diagram. Then, locate several additional points that satisfy the condition and plot them as well. Plot enough points so that a pattern (a shape, a path) is starting to appear.

4. Through these plotted points draw a dotted line to indicate the locus (or path) of the points.

5. Describe in words the geometric path that appears to be the locus.

Describe the following Locus: 1. the locus of points 1 cm from a given line, m.

Consider: 2. Niko gets a stone wedged in the tire of his dirt bike. He lifts the bike and spins the wheel to look for the stone. Describe the path of the stone as Niko spins the wheel?

Ponder: 3. If Niko "pushed" the bike along the ground to search for the stone, would the path of the stone still be a circle?

Day 2 – More Basic Loci Date: ___________ Warm Up: 1. The locus of points equidistant from two fixed points is ... (a) one circle (b) one straight line (c) two circles (d) two straight lines

2. What is the locus of points 3 inches from point B?

(a) a line (b) a circle (c) a triangle

3. The locus of points equidistant from two parallel lines is ...

(a) one circle (b) one straight line (c) two circles (d) two straight lines

4. The locus of points equidistant from two intersecting lines is ...

(a) one circle (b) two parallel lines (c) two circles (d) a pair of angle bisectors

5. The locus of points at a given distance from a straight line is...

(a) one straight line (b) a circle (c) two parallel lines (d) two intersecting lines

Let’s practice some application problems of Locus . . .

1. Describe the locus of points equidistant from two concentric circles having radii of 7 cm and 11 cm respectively.

2. Describe the locus of all of the possible vertices of isosceles triangles that have the same base. 3. Describe the locus of the centers of circles tangent to each of two parallel lines that are 10 inches apart.

4. The student radio station has a broadcasting range of 24 miles. Describe the locus of points which represents the outer edge of the broadcasting range.

5. A straight road is 8 feet wide. A gardener is planning to plant flowers 6 feet from the center line of the road. Describe where the flowers will be planted.

6. Ben skis through a park that is bounded on two sides by straight intersecting streets. Ben skis so that he is always the same distance from each street. Describe Ben's path.

7. There are two buoys in a lake. A scuba diver swims down beneath the water so that he is always equidistant from both buoys. Describe his path.

8. Describe the locus of the center of the wheel of a train that is moving along a straight, level track.

9. Describe the locus of a car that is driven down a straight road equidistant from the two opposite parallel curbs on the side of the road?

center of

wheel

Day 3 – Locus in the Coordinate Plane Date: ___________ Warm Up: Graph the following equations on the coordinate grids below. Graph 1: x = 3 an d y = -2 Graph 2: (x + 1)2 + (y - 3)2= 4 Graph 3: y = x and y = -x

Locus and Coordinate Geometry

Review of Coordinate Terms/Equations:

General Equation of a circle: __________________________________________________

Vertical lines are ____________ equations

Horizontal lines are _________ equations

Abscissa - ______________________________________________________________________

Ordinate - ______________________________________________________________________ Basic Loci: 1. The locus of points whose abscissas are the constant a is the line parallel to the y-axis

and whose equation is x = a.

The locus of points whose abscissa is 12 is ____________________________ The locus of points 3 units from x = 12 is ____________________________

2. The locus of points whose ordinates are the constant b is the line that is parallel to the

x-axis and whose equation is y = b.

The locus of points whose ordinate is –3 is __________________________ The locus of points 5 units from y = -3 is ____________________________

3. The locus of points which form a straight line can be written in the form y = mx + b.

The locus of points whose ordinates are 4 more than three times the abscissa is:

The locus of points whose abscissas are 8 less than four times the ordinate is:

4. The locus of points equidistant from one given point is a circle

The locus of points 5 units from the origin is __________________________________

The locus of points 2 units from (-3, 4) is ______________________________________

Let’s Practice:

1. True or False: The locus of points equidistant from the lines x = -1.5 and x = 1.5 is the x-axis.

2. What is the equation of the locus of points equidistant from the lines y = -2 and y = 3?

(a) y = 0 (b) y = 1/2 (c) x = 1

3. What is the equation of the locus of points 5 units away from the y-axis?

(a) y = 5 (b) x = 5 or x = -5 (c) y = 5 or y = -5

4. True or False: The locus of points equidistant from the line x = 2 could be two lines whose equations are x = 0 and x = 5.

5. What is the equation of the locus of points equidistant from the points (4,2) and (-2,2)? (a) y = 1

(b) x = 1 (c) x = -1

6. What is the equation of the locus of points equidistant from the x-axis and the y-axis?

(a) y = x and y = -x (b) y = 2 and y = -2 (c) x = 1 and x = -1

7. Write the equation and graph the locus of points that are 3 units from the point (1, -2). ***FOR GRAPHS – be sure to graph any given information and locus graph should be DOTTED

Independent Practice: Find the equation of the locus that satisfies each of the following conditions: (Use graph paper if necessary):

1. Write the equation of the locus of

points in the coordinate plane that

are 7 units from the point (-4, 9).

2. Write the equation of the locus of

points that are 4 units from the x-

axis.

3. Write the equation of the locus of

points that are 5 units from the line

x = 3.

4. Write the equation of the locus of

points that are 2 units from the line

y = -1.

5. Write the equation of all points

whose ordinates are one-half as

great as their abscissas.

6. Write the equation of all points the

sum of whose ordinates and

abscissas is –4.

7. Graph and write the equation of the

locus of points equidistant from the

points (5, -4) and (-1, -4).

8. Write the equation of all points

whose abscissas exceed twice their

ordinates by 3.

9. Write the equation of all points

equidistant from the lines whose

equation are y = x and y = x – 4.

10. Write the equation of the locus of

points in the coordinate plane that are

7 units from the origin.

Day 4 - Compound Loci

Warm Up:

1. Write the equation(s) of the locus of points that are 3 units from the line x = -1. 2. Write the equation(s) of the locus of points that are 6 units from the point (3,-1) 3. Write the equation(s) of the locus of points that is equidistant from the lines x=0 and y=0. A compound locus problem involves two, or possibly 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". Intersections of Loci: To find points that satisfy two or more conditions:

1. Draw the locus of points that satisfy the first condition. 2. Draw the locus of points that satisfy the second condition. 3. Locate the points of intersection of these loci and circle or place an X on them. These

points are the points that satisfy both conditions. Ex 1: Parallel lines l and m are 2 cm apart and A is a point on line l. What is the locus of points equidistant from l and m and 3 cm from A? Ex 2: A and B are points on a given line l and P is a point not on l. What is the locus of points equidistant from A, B and P?

Ex 3: How many points are equidistant from points A and B and also 4 units from the segment AB?

Ex 4: How many points are equidistant from two intersecting lines and also 3 inches from their point of intersection?

Real-Life Applications Ex 1: Nastaja’s backyard has two trees that are 40 feet apart, as shown in the diagram. She wants to place lampposts so that the posts are 30 feet from both of the trees. Draw a sketch to show where the lampposts could be placed in relation to the trees. How many locations for the lamppost are possible? Ex 2: Matt has a treasure map, represented in the diagram, that shows two trees 8 feet apart and a straight fence connecting them. The map states that treasure is buries 3 feet from the fence and equidistant from two trees.

a) Sketch a diagram to show all the places where the treasure could be buried. Clearly indicate in your diagram where the treasure could be buried.

b) What is the distance between the treasure and one of the trees?

Day 5 - Compound Loci

Warm Up:

a. Draw and write the equation of the locus of points 4 units

from the point (1, 4)

b. Draw and write the equation of the locus of points 3 units

from the y-axis

c. How many points satisfy both conditions stated in parts a

and b?

Problem 1: In the diagram below, town C lies on straight road p. Sketch the points that are 6 miles from town C. Then sketch the points that are 3 miles from road p. How many points satisfy both conditions?

Problem 2: A triangular park is formed by the intersection of three streets, Bridge Street, Harbor Place, and College Avenue, as shown in the accompanying diagram. A walkway parallel to Harbor Place goes through the park. A time capsule has been buried in the park in a location that is equidistant from Bridge Street and College Avenue and 5 yards from the walkway. Indicate on the diagram with an X each possible location where the time capsule could be buried.

Problem 3: 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.

Problem 4: 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.

5. What is the number of points in a plane two units from a given line and three units from a given point on the line?

(1) 1 (2) 2 (3) 3 (4) 4

6. Graph the line y = -6.

a. Graph the locus of points that are 3 units away from the point (1, -1). Write the equation of this locus.

b. Describe the locus of points that are h units away from the line y = -6. Then, write the equation(s) of this locus.

c. How many points of intersection satisfy the conditions in parts A and b if

h = 1

h = 8

h = 5

Give the coordinates for each part above.

7. Two points A and B are 6 units apart. How many points are there that are equidistant from both A and B and also 5 units from A?

(1) 0 (2) 1 (3) 2 (4) 3

8. Parallel lines r and s are 8 meters apart, and A is a point on line s. How many points are equidistant from r and s and also 4 meters from A?

(1) 0 (2) 1 (3) 2 (4) 3

9. A given point P is 10 units from a given line. How many points are 3 units from the line and 5 units from point P?

(1) 0 (2) 1 (3) 2 (4) 3

10. Two points A and B are 7 units apart. How many points are there that are 12 units from A and also 4 units from B?

(1) 0 (2) 1 (3) 2 (4) 3

11. a. Draw the locus of points equidistant from the points (4,1) and (4,5) and write the equation for this locus.

b. Draw the locus of points equidistant from the points (3,2) and (-4,2) and write the 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.

12. a. Draw the locus of points 3 units from the y-axis and write the equation for this locus.

b. Draw the locus of points 4 units from the

origin and write the equation for this locus.

c. How many points that satisfy both conditions stated in parts a and b?