Last Packet Geometry

Locus

Constructions

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Locus WS 1 Name __________________________

Geometry Date ______________ Block ________

The set of all points that satisfy a condition is called a lOCUS. Draw your own diagram

to illustrate each locus rule below.

Locus rule 1:

The set of points equidistant from a given point:

Locus rule 2:

The set of points equidistant from a given line:

Locus rule 3:

The set of points equidistant from 2 parallel:

Locus rule 4:

The set of points equidistant from the sides of an angle:

Locus rule 5:

The set of points equidistant from two intersecting lines:

Locus rule 6:

The set of points equidistant from 2 given points:

Locus rule 7:

The set of points equidistant from a line and a point not on the line:

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Locus of Points and Five Fundamental Loci

Locus: the set of all points that satisfy a given condition or set of conditions

Locus Theorem 1: (point)

The locus of points at a fixed distance, d, from point P is a circle

with the given point P as its center and d as its radius.

Consider: When he is not in the house, Fido is tied to a stake in the backyard. His leash, attached to

the stake, is 15 feet long. When traveling at the end of his leash, what is the locus of Fido's path?

Point P, from the theorem, is the stake

to which Fido, the dog, is tied. His

leash is 15 feet long. The path that Fido

can travel at the end of his leash is "the

locus of points".

The locus of points at a distance of 15

feet from point P is a circle (with center

P and radius 15).

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Locus Theorem 2: (line)

The locus of points at a fixed distance, d, from a line, l, is a pair of parallel lines d

distance from l and on either side of l.

Consider: Your teacher has placed a strip of tape on the classroom floor which forms a straight

line. The teacher gives each student a yard stick and asks that each student stand exactly 3 feet away

from the line on the floor. Can you picture what will happen? If you, and all of your classmates, stand

exactly 3 feet away from the line, describe where you and your classmates will be standing.

Answer:

You and your classmates will form two straight lines on either side of the tape on the floor,

at a distance of 3 feet away from the tape.

You and your classmates are the locus of points equally distant (equidistant) from a given

line (the tape on the floor).

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Locus Theorem 3: (two points)

The locus of points equidistant from two points, P and Q, is the perpendicular

bisector of the line segment determined by the two points.

Consider: You are playing a game of paint ball. Two of your friends are hiding behind trees that are 10

feet apart. Where could you possibly stand so that your firing distance to each friend is exactly the same

length?

Answer:

At first it may seem that there is only ONE spot to stand

where you are the same distance from both of your friends - that spot being directly between your

friends, 5 feet from each friend.

But, as the diagram shows, there are actually many spots that will

position you exactly the same distance from both friends. Notice the formation of the isosceles

triangles, where the congruent (equal) sides represent the distances to each friend.

The different positions where you might stand form the locus of points equidistant (equally distant) from

your two friends. This line is the perpendicular bisector of the segment joining your two friends.

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Locus Theorem 4: (parallel lines)

The locus of points equidistant from two parallel lines, l1 and l2 , is a line parallel

to both l1 and l2 and midway between them.

Consider: During your morning jog, you run down an alley between two buildings which are parallel to

one another and are 20 feet apart. Describe your path through the alley so that you are always the same

distance from both buildings.

Answer:

To maintain an equal distance from each building, you must jog in a straight line parallel to the

buildings and halfway between them. In this problem, since the buildings are 20 feet apart, you will jog

on a line 10 feet from each building.

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Locus Theorem 5: (intersecting lines)

The locus of points equidistant from two intersecting lines, l1 and l2, is a pair of

bisectors that bisect the angles formed by l1 and l2 .

Consider: Indiana Jones (Indy) is searching for a treasure. His treasure map shows a V-shaped canyon

formed by two vertical cliffs. The directions state:

"Start at the point where the cliffs converge. Walk through the canyon staying an equal distance from

each cliff......"

Can you picture the path Indy needs to take to satisfy these directions?

Description:

The red line represents the locus which is equidistant from the two cliffs. This line is the bisector of the

angle formed by the two cliffs. Any point on this line, when measured as shown by the green lines, is

the same distance from each cliff.

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What is the locus of points equidistant from the endpoints of a given line

segment?

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Construct the locus of points in the interior of an angle equidistant from the

rays that form the sides of the given angle.

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Locus in the Coordinate Plane

Give an equation for the locus of points:

1. 3 units to the right of the y-axis 2. 5 units to the left of the y-axis

3. 7 units below the x-axis 4. 20 units above the x-axis

5. equidistant from the points (-6, 0) and (6, 0)

6. equidistant from the points (0, 3) and (0, -3)

______________________________________________________________________

Describe the following as a locus of points in relation to the x-axis or y-axis.

7. the line with equation x = -6 8. the line with equation y = 5

9. the line with equation y = -2 10. the line with equation x = 8

_______________________________________________________________________

Write the equation(s) for the locus of points with the given criteria.

11. 1 unit below the line with equation y = 3

12. 2 units to the right of the line x = -5

13. 3 units from the line with equation x = 6

14. 2 units from the line with equation y = -1

15. 1.5 units from the line with equation y = 0.5

16. 5.5 units from the line with equation x = -2.5

17. equidistant from y = -6 and y = 4

18. equidistant from the lines with equations x = 5 and x = -2

19. equidistant from the lines with equations x = -2 and x = 1

20. equidistant from the lines with equations y = -3 and y = -8

_______________________________________________________________________

Write the equation for the locus of points that are equidistant from the given points.

21. (1, 0) and (5, 0) 22. (3, -1) and (3, 7)

23. (-4, 1) and (-4, 4) 24. (-5, 3) and (0, 3)

_______________________________________________________________________

Determine the number of points in the given locus. Show your work on graph paper or

in a sketch with lengths clearly marked!

25. 5 units from the origin and 3 units from the x-axis

26. 13 units from the origin and 5 units from the x-axis

27. 2 units from the origin and 4 units from the x-axis

28. 1 unit from the origin and 1 unit from the y-axis

29. 2 units from the origin and 1 unit from the line with equation x = 2

30. 3 units from the origin and 2 units from the line with equation x = -5

31. equidistant from the x-axis and the y-axis and 4 units from the origin

32. equidistant from the lines y = x and y = -x and 2 units from the origin

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Locus in a Plane

Sketch each locus or compound locus set with lengths clearly marked.

33. Given point P on AB . What is the total number of points that are a distance of 4

cm from AB and also a distance of 10 cm from point P?

34. What is the total number of points that are equidistant from two intersecting lines

m and n and also 3 inches from line m?

35. Point P is 2 inches from CD . What is the total number of points that are 1 inch from

CD and also 1 inch from P?

36. Parallel lines AB and CD are 6” apart and point P is on AB . What is the total

number of points that are equidistant from AB and CD and also 6” from P?

37. Points A and B are 5 in. apart. What is the total number of points that are 2 in. from

A and also 4 in. from B?

38. Points R and S are 4 cm apart. How many points are equidistant from R and S and

4 units from point S?

39. Point P is 2 in. from AB . What is the total number of points 4 in. from AB and also 6

in. from P?

40. Parallel lines m and n are 10 in. apart. Point C lies between m and n at a distance

of 3 inches from m. What is the number of points that are equidistant from m and n

and also 7 inches from C?

41. How many points in the interior of <A are equidistant from the sides of <A and also

2 inches from point A?

42. Given point P is 3 inches from line m. How many points are 1 inch from line m and

4 inches from point P?

43. What is the total number of points equidistant from two intersecting lines and 1

centimeter from their point of intersection?

a) 1 b) 2 c) 3 d) 4

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Locus WS 2 Name __________________________

Geometry Date ______________ Block ________

1. Point P is 8 cm from AB . How many points are 6 cm from point P and also 3 cm from

AB ?

2. Parallel lines l and m are 4 cm apart and P is a point on line l. What is the total

number of points that are equidistant from l and m and also 2 cm from P?

3. Points A and B are 4 in. apart. What it the total number of points equidistant from

both A and B and also 3 in. from A?

4. Given M is the midpoint of AB and AB = 6. How many points are equidistant from

A and B, and 4 units from AB , and 5 units from point A?

5. What is the total number of points that are equidistant from two intersecting lines

and also 5 inches from their point of intersection?

6. Point A is 9 in. form a given line. How many points are both 4 in. from this line and 6

in. from A?

7. Two points A and B are 7 in. apart. How many points are 10 in. from A and also 3 in.

from B?

8. Points P and Q are 7 units apart. How many points are 4 units from P and 4 units

from Q?

____________________________________________________________________

Write the equation(s) for the locus set which satisfy the given criteria. Sketch or graph

the given information and the locus set to justify your answers.

9. The locus of points equidistant from the points A(3, 8) and B(3, -2).

10. The locus of points 3 units from the point (-4, -1).

11. The locus of points equidistant from the x-axis and the y-axis.

12. The locus of points 5 units from the line x = 2.

13. The locus of points equidistant from the lines with the equations y = 3 and y = 9.

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Leave all your construction lines on your diagrams.

1. David lives 4km away from the school and 6km away from his friend Andrew’s house. Find two

possible places for his house. Use a scale of 1cm = 1km. 2. Kelly has an apple tree in her garden. It is 3m away from her swing, and 500cm

away from the fence at the edge of her garden. Show two places where her tree could be. Use a scale of 2cm = 1m.

School

٭ Andrew’s house

٭

٭ Swing

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3. Tom wants to put a TV into his bedroom.

He must place it at least ½ metre away from the window, because of the light, and at least 1

metre away from the adjoining wall so that the noise does not disturb his parents in the next room. It must be at least 2½ metres away from his chair, otherwise it will be bad for his eyes, but it must also be less than 3 metres away from the socket on the wall as this is how long the lead for his TV is.

Tom wants to know where he can put his TV. By shading the areas that his TV cannot go, show the possible places that Tom could put his

TV. Label this area clearly Use a scale of 1cm to 500 cm.

٭ Chair

Window

Wall

Socket ٭

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Constructions

A B

C

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A

16

17

18

19

20

21

22

23

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_______________________________________________________________________

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27

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A B

A B

C

A B

C

Constructions WS1 Name ___________________________

Geometry Date ___________ Block _________

1. Construct the perpendicular bisector of the given segment.

2. Construct the bisector of the given angle.

3. Construct a line parallel to the given line which passes through the given point.

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A B

A B

C

A B

C

4. Construct an equilateral triangle with a side length congruent to the given

segment.

5. Construct a line which is perpendicular to the given line and which passes through

the given point.

6. Construct an angle congruent to the given angle.

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Constructions WS2 Name ____________________

Geometry Date _________ Block ______

_________________________________________________________________

_________________________________________________________________

__________________________________________________________________

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5.

_________________________________________________________________

6.

___________________________________________________________________

7.

____________________________________________________________________

8.

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Concurrent Lines Points of intersection in a triangle

Centroid- The point of intersection of all the medians of a triangle. The centroid is

always inside the triangle. From the vertex of a triangle to the centroid is

always equal to 2

3 the length of the median.

Incenter- The point of intersection of all the angle bisectors of a triangle. This point is

the center of the inscribed circle of the triangle. This point is always in the triangle as

well.

Orthocenter- The intersection of the altitudes of a triangle is called the orthocenter.

The orthocenter can be on the outside of the triangle (obtuse triangles) or on the

triangle itself (right triangle).

Circumcenter- The intersection of the

perpendicular bisectors of the sides of a triangle

is called the circumcenter. This is the

center point of the circumscribing circle.

It can also be inside, outside or

on the circle just like the orthocenter.

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Point of Concurrency Worksheet Give the name the point of concurrency for each of the following.

1. Angle Bisectors of a Triangle ________________________

2. Medians of a Triangle ___________________________

3. Altitudes of a Triangle ___________________________

4. Perpendicular Bisectors of a Triangle __________________________

Complete each of the following statements.

5. The incenter of a triangle is equidistant from the _______________ of the

triangle.

6. The circumcenter of a triangle is equidistant from the _________________ of

the triangle.

7. The centroid is ___________ of the distance from each vertex to the midpoint

of the opposite side.

8. To inscribe a circle about a triangle, you use the ____________________

9. To circumscribe a circle about a triangle, you use the ________________

10. Complete the following chart. Write if the point of concurrency is inside,

outside, or on the triangle.

Acute Obtuse Right

Circumcenter

Incenter

Centroid

Orthocenter

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In the diagram, the perpendicular bisectors (shown with dashed segments) of ABC

meet at point G--the circumcenter. and are shown dashed. Find the indicated

measure.

11. AG = __________ 12. BD = __________

13. CF = __________ 14. AB = __________

15. CE = __________ 16. AC = __________

17. mADG = __________

18. IF BG = (2x – 15), find x.

x = __________

In the diagram, the perpendicular bisectors (shown with dashed segments) of MNP

meet at point O—the circumcenter. Find the

indicated measure.

19. MO = ___________ 20. PR = __________

21. MN = __________ 22. SP = __________

23. mMQO = __________

24. If OP = 2x, find x.

x = __________

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Point T is the incenter of PQR.

25. If Point T is the incenter, then Point T

is the point of concurrency of

the ___________________________.

26. ST = __________

27. If TU = (2x – 1), find x.

x = __________

28. If mPRT = 24º, then mQRT = __________

29. If mRPQ = 62º, then mRPT = __________

Point G is the centroid of ∆ ABC, AD = 8, AG = 10, BE = 10, AC = 16 and

CD = 18. Find the length of each segment.

30. If Point G is the centroid, then Point T

is the point of concurrency of

the ___________________________.

31. DB = __________ 32. EA = __________

33. CG = __________ 34. BA = __________

35. GE = __________ 36. GD = __________

37. BC = __________ 38. AF = __________

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Point S is the centroid of RTW, RS = 4, VW = 6, and TV= 9. Find the length of each

segment.

39. RV = __________

40. SU = __________

41. RU = __________

42. RW = __________

43. TS = __________

44. SV = __________

Point G is the centroid of ∆ ABC. Use the given information to find the value of the

variable.

45. FG = x + 8 and GA = 6x – 4

x = __________

46. If CG = 3y + 7 and CE = 6y

y = __________