Geometry Chapter 9 Circle Vocabulary Arc Length Angle & Segment Theorems with Circles Proofs

1

Chapter 9: Circles

Date Due

Section

Topics

Assignment

Written Exercises

9.1

Definitions Worksheet (pg330 classroom ex.all) Pg. 331 #4, 6, 7, 12-15, 17 Pg.337 #1-3 (mixed review-bottom of page)

9.2

Tangents Circumscribed vs. Inscribed Common Tangent Tangent Circles

Pg. 335(bottom)-337 #1-7, 8(not d), 10, 14-20 even

9.3

Arcs (minor and major) Central <’s Arc Addition Postulate Congruent Arcs Length of an Arc

Pg. 341 (bottom)- 342 # 1-6, 10, [note m<CAO = m<2] , #16

9.4

Arcs and Chord Relationships

Pg. 347 # 1-9, 12, 18, 22

9.5 Inscribed Angles Pg. 354-355 #2-8, 19-21,

9.6 Angles formed by Chords,

Tangents and Secants Pg. 359-361 #1-10, 12-24 even

9.7

Lengths of Segments in a Circle

Pg. 364 (bottom)- 366 #2-8 even, 14-22 even

More Proofs

Proofs Worksheet

Review Study For Test

Chapter 9

Extra Practice Pg.349 (Self Test 1) #1-6 Pg.367 (Self Test 2) #1-8 Pg.369-370 (Chpt Rev) #1-24 Pg.371 (Chpt Test) #1-18

2

Circle Introductory Vocabulary Name _______________________

Geometry Date ___________ Block _______

Use appropriate notation to name the following in the given diagram.

Write a short explanation or definition as needed.

circle:

center:

diameter:

radius:

chord:

arc:

semicircle:

major arc:

minor arc:

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secant:

tangent:

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inscribed polygon:

circumscribed polygon:

3

p.330 Class Exercises

1. Name three radii of O .

2. Name a diameter.

3. Consider RS and RS . Which is a

chord and which is a secant?

4. Why is TK not a chord?

5. Name a tangent to O .

6. What name is given to point L?

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7. Name a line tangent to sphere Q.

8. Name a secant of the sphere and

a chord of the sphere.

9. Name 4 radii. (none are drawn

on the diagram)

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10. What is the diameter of a circle with radius 8? 5.2? 4 3 ? j?

11. What is the radius of a sphere with diameter 14? 13? 5.6? 6n?

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The radius of circle O has a length of 20. Radii OA and OB are drawn in,

forming an angle with the given measure. Find the length of AB using

your knowledge of isosceles and special triangles.

a) m<AOB = 90 b) m<AOB = 60 c) m<AOB = 120

4

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9-3: Arcs and Central <'s

& 11-6: Arc Length

An arc is measured in degrees - Its measure is equal in measure of the

central angle which intercepts it.

Arcs are ≅ iff. their central <'s are ≅, with angles 0<θ<360.

5

The central angles are equal in measure....

While the arcs are equal in measure, the arcs are different in length!

Arc length is related to circumference....

C = πd or  C = 2 π r

Arc length....

l = 360

central measured

l = 2360

central measurer

Think about it -- arc length is a fractional part of the circumference of the

circle & the circle is 360 degrees!!!!

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Thm: the measure of a central angle = the measure of the intercepted arc

6

Central Angle & Arcs Notes Name _________________________

Geometry Date ________ Block _________

Find the measure of each arc in the diagram.

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Use the diagram to answer the following:

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7

Arc Length Practice WS

Determine the length of an arc with the given central angle measure,

m<M, in a circle with radius r. Give your answers in simplest form in terms

of .

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Determine the length of an arc with the given central angle measure,

m<M, in a circle with radius r. Give your answers rounded to the nearest

hundredth.

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Determine the degree measure of an arc with the given length, L, in a

circle with radius r. Give your answers rounded to the nearest tenth.

Extra Practice: p.341 CE(1-13)

8

Thm: A line tangent to a circle the line is perpendicular to the radius @

the pt of intersection

Converse: A line which is

perpendicular to the radius @

a point on the circle the

line is tangent to the circle

* the circle & line must be

Coplanar!

Thm: parallel lines/chords in a circle intercepted arcs are congruent

Thm: Tangent segments from an external point are congruent

Thm: If a line in the plane of a circle is perpendicular to the radius

(diameter) at its outer endpoint, then the line is tangent to the circle.

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9

3. If AP = 12 and BP = 6, find AO.

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Tangents with Circles

Internal Tangent Line External Tangent Line Tangent Circles

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10

Thm: In the same circle or congruent circles,

congruent arcs congruent chords

Thm: If a diameter of a circle is perpendicular to a chord it bisects both

the chord and its intercepted arc.

Converse: if diameter bisects a

chord it is perpendicular to the

chord @ its midpoint

Thm: In the same circle or congruent circles,

If 2 chords are equidistant from the center the chords are congruent.

Converse: if chords are congruent

the chords are equidistant from

the center

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11

Classwork:

1) p.349 ST1 (1-6)

2) p.346 Class Exercises (1-6)

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Thm: the measure of an inscribed angle = half the measure of the

intercepted arc

* 2 inscribed angles which

intercept the same arc

angles are congruent

* An angle inscribed in a

semicircle right angle

Proof of theorem on next page…

12

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Corollary 1:

2 inscribed angles which intercept the same arc are congruent.

13

Corollary 2:

An angle inscribed in a semicircle is a right angle.

Corollary 3: quadrilateral inscribed in a circle opposite angles are

supplementary

Thm: an angle formed by a tangent & a chord its measure = half the

measure of the intercepted arc

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Examples (p.353 #4 – 9) Find the value for x and y in each question.

14

Inscribed Angles Notes Name ___________________________

Geometry Date __________ Block ___________

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15

Angles formed by a tangent and a chord – Notes

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16

Thm: If 2 chords intersect inside a circle the products of the segments

on each chord are equal

Proof of theorem:

17

Thm: an angle formed by 2 secants/tangents its measure = half the

difference of the measures of the intercepted arcs

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p.359 CE(1-10)

6

18

Other Angle Relationships Name _______________________

Geometry Date _____________ Block _____

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19

20

Circle Formulas Name______________________

Geometry Date __________ Block _____

For each of the given diagrams, fill in the appropriate formulas for the

angle measures and lengths of the segments.

Circle with Center O

m QOP

m QRP

Diameter AB and tangent BD

m ACB

m ABD

m QUR

Segment Relationship:

HJ and LJ are tangents

m JHL

m HJL

Segment Relationship:

OP

R

Q

AO

B

DC

U

O

Q

T

R

S

J

O

L

H

K

21

Secants UXV and UYW

m VUW

Segment Relationship:

Secant MRN and tangent MP

m PMN

Segment Relationship:

BD OC

Segment Relationship:

Arc Relationship:

m FG

length of FG

Y

X

O

W

U

V

R

O

P

M

N

D

BO

C

E

O

G

F

22

Review for quiz WS1 Name _____________________________

Geometry -- chapter 9 Date ______________ Block _________

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31

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32

Review for Quiz WS2 Name ____________________________

Geometry – chapter 9 Date ______________ Block ______

In Circle E, m BD=200, m DF =1800, and m AF =450. GA is tangent to

circle E at A. Find the following:

1. m<CAG = _______

2. m<GCA = _______

3. m<CGA = _______

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In the figure, XY is tangent to circle Z at X.

4. If m XW = 950, find m<YXW. _______

5. If m<YXW = 1000, find m XW . _______

6. If m XW = x + 15, find m<YXW in terms of x. _______

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In Circle P, m<LPJ = 300 and m<KMJ = 450. Find the following:

7. m<LMP = _______

8. m<JPK = _______

9. m<MJK = _______

10. m<LPM = _______

11. m MLJ = _______

12. m<MPK = _______

13. m<JPK = _______

14. m LJ = _______

15. m KM = _______

16. m<PLM = _______

33

In Circle J, JP KL at S.

17. SL _______

For question 18 & 19,

give answers in simplest radical form!

18. IF JL = 4, and JS = 1,

What is KS? _______ What is KL? _______

19. IF JK = 26 and JS = 11,

What is KS? _______ What is KL? _______

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Determine the length of an arc with the given central angle measure,

m<P, in a circle with radius r. Give answers in terms of and then

rounded to the nearest tenth.

20. m<P = 400; r = 6 21. m>P = 200; r = 8

22. m>P = 1180; r = 30 23. m>P = 1300; r = 61

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Determine the degree measure of an arc with the given length, L, in a

circle with radius r. Give answers rounded to the nearest degree.

24. L = 27; r = 5 25. L = 100; r = 79

26. L = 35; r = 11 27. L = 2.3; r = 85

34

Thm: If 2 chords intersect inside a circle the products of the segments

on each chord are equal

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35

Thm: If 2 secants are drawn to a circle from an external point the

products of the external secant and the whole secant are equal

Proof of theorem:

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36

Thm: If a secant segment and a tangent segment are drawn to a circle

from an external point the product of the secant segment and its

external segment is equal to the square of the tangent segment

Proof of theorem:

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37

B

D

C

A

CB

P

A

D

T

B

P

A

H

F

G

J

O

K

Segments in Circles WS Name _____________________

Geometry Date __________ Block _____

1. Chords AC and BD intersect at point E.

a) If AE = 5, AC = 13, and DE = 10, find BE.

b) If AE = 3, CE = 4x+1, DE = 9, and BE = 2x-1, find x.

c) If AC bisects BD , AE = 8, and EC = 32, find BD.

2. In Circle O, diameter HJ is perpendicular to

chord FG at K. If HO = 13 and FG = 10, how far

is the chord from the center of the circle?

3. In Circle O, tangent PT and secant PBA

intersect at point P, outside the circle.

a) If PT = x, PB = 3, and AB = 13, find x.

b) If PT = 3, PB = x, and AB = 8, find x.

4. Secants PBA and PCD intersect at point P,

outside the circle.

a) If PB = 8, PA = 18, PC = 9, and PD = x, find x.

b) If PB = 4, AB = 17, PC = x, and CD = 5, find x.

c) If PB = 6, AB = 9, PC = 8, and PD = x, find x.

38

Circle Proofs WS 1 Name ___________________

Geometry Date _______ Block _______

1. Given: Circle O with diameter AOD ,

tangent CA , 2mAB mAE

Prove: ADOA = OCBD

2. Given: Circle O, tangents PR and PV

Prove: RPO VPO

3. Given: CT CH

Prove: HA TD

4. Given: chords AB and CD of circle O intersect at E, an interior point of

circle O;

chords AD and CB are drawn.

Prove: (AE)(EB) = (CE)(ED)

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5) 6)

7) 8)

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9)

40

Circle Proof WS 2 Name __________________

Geometry Date ___________ Block _______

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42

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43

x

3

4

10

80 o

120 o

90 o

x

D

C

BA

AB

x

280 o

xB

A

70 o

O

310 o

A

B

x

O

100 o

x

B

A

C

A

Bx

70 o

O

G100 o

x

E

H

F40 o

O

x

D

C

B

60 o

OC

B

A

y

x

100 o

O

x

y

A

B

C

3y

O

DC

BA

x70 o O

80 o

x

A B

C DO D

C

BA

x

30 oO

80 o

x

A B

C D

120 o

O

100 o

70 o

x

A B

C D

O

x

260 o

O

B

A

T

80 o

x

O

D

A

T

2y

x

y

O

E

A

T

4y

y x

O

G

A

T

x

70 o

O

H

A

T

x

D

C

B

A

100 o

40 o O

100 o

A

B

C D

x

O

40 o

A B

C

D

x

O

2y

y

A B

C

Dx

O

4

2

|----------x------------|

Review Chapter 9 WS 1 Name _____________________________

Geometry Date ___________ Block _________

1. 2. 3.

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4. 5. 6. 7. 8.

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9. 10. 11. 12. 13.

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14. 15. 16. 17. 18.

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19. 20. 21. 22. 23.

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24. 25. 26. 27. 28.

y

x

R T

A

DS

B

E

C

8

50 o A

B

C

D

x

O 1000

1400

44

x

100 o

60 o

DB

O

C

A

E

5y

x

2y

yy

O

260 o

x

O

3y

x

y

O

3y

y

x

O

F

E

DC

BA

4

32

1

O

12

E

x

D

C

B

A68

O 16

4

EA

B

C

DFind AB

O

A B

C

AB = 8

Radius = ?

2

O x

PD = 12

45

C

AP

B

D

radius = 4

x

3

O50 ox

O x 60 oO

60 o

x

O

A

C

B x

m DFE = 170o

O

D

E

F

G

29. 30. 31. 32. 33.

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34. given: AB is tangent; AC is secant; , ,BC DE FC

are chords; 50mEB ; 4 50mBC x ;

mCD x ; 25mDF x ; 15mFE x

Find:

mBC 1m

mCD 2m

mDF 3m

mFE 4m

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35. 36. 37. 38.

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39. 40. 41.

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42. 43. 44.

In circle O, radii ,OA OB ,

and chord AB are drawn.

If OA = 2x+8, OB = x+24, and

AB = 3x-8, find OA, AB, and

m<AOB.

45

S

J

P

K L

K

J

P

M

L

B

DF E

A

GC

16

11

x x

416

12

x

7

6

4

Circles Chapter Review WS 2 Name _______________________

Geometry Date __________ Block ______

NOTE: Diagrams may not be to scale!!!!!!

1. Find the arc length for Circle P with radius 6 if m<P = 40 (nearest tenth).

2. Find the measure of the central angle that intercepts an arc of length

27 on a circle with a radius of 5 (nearest degree).

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3. If JP KL at S, JK = 26, and JS = 11, find:

KS = _____

KL = _____

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4. If m<LPJ = 30 and m<KMJ = 45, find:

JK = ______

MK = ______

LM = ______

m<LMP = ______

m<PLM = ______

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5. If 20, 180, 45mBD mDF mAF , find:

m<CAG = ______

m<GCA = ______

m<CGA = ______

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Find the value of x. Show algebra for questions 6 - 25.

6. 7. 8.

46

x

3

6

x

250o

100o

50o

x

x145 o

x

1510

x

63 o

130 o

70 o

x

x+3

6

4x

x

T

S

R

3x+9

54

x

87o

94o

9. 10. 11.

12. 13. 14.

15. 16. (nearest tenth) 17. (nearest tenth)

18. 19. 20.

4 15

5 15

m TSR x

m RTS x

x+3

2x

4

47

x26 o

87 o 92o3x-95x+9 2

1

4x

5x

3x

R

P

Q

D

O

A

F

B

C

E

21. 22. 23.

1 2.5

2 1.5 14

m x

m x

24.

Find m R .

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26. In Circle O, FA is tangent, FEDB is a secant, ADC and AB are

chords, m CE = 40, m AB = 130, and m<CAB = 60.

a) m BC = _____

b) m<EBA = _____

c) m<ADE = _____

d) m<F = _____

e) m<FAC = _____

48

E

F

D

O

A

CB

E

B

D

O

P

C

G

F

A

E

B

A

O

D

P

C

27. In the accompanying diagram of Circle O with inscribed isosceles

triangle ABC, AB AC , m BC = 60, FC is a tangent and secant FBA

intersects diameter CD at E.

a) m<ABC = _____

b) m AD = _____

c) m<DEB = _____

d) m<AFC = _____

e) m<BCF = _____

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28. In the accompanying diagram of Circle O, secant ABP , secant CDP ,

and chord AC is drawn; chords AD and BC intersect at E, tangent GCF

intersects circle O at C, and m AB : m BD : m DC : mCA = 8:2:5:3.

a) mCA = ______

b) m<ACB = ______

c) m<P = ______

d) m<AEB = ______

e) m<DCF = ______

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29. In the accompanying diagram of Circle O, AOED is a diameter, PD is

a tangent, PBA is a secant, chords BD and BEC are drawn, m<DAB = 43,

and m<DEC = 72.

a) m<BDP = ______

b) m AB = ______

c) m AC = ______

d) m<P = ______

e) m<CBD = _____

49

Chapter 9--Theorems/Corollaries/Postulates

Formulas to know: 2A r C d

length of arc: 360

xl d ; x = measure of central

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Hints: draw radii to endpts. of a chord [look for special right s]

find isosceles s formed w/ radii and a chord

find right s formed w/ tangent [radii or diameter a side of the ]

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Basics 1. line tangent to line is to radius @ pt. of tangency

2. [coplanar line & ] line to radius @ pt. on line tangent to

3. tangents to from exterior pt. are

4. [in 1 or s] arcs chords

5. diameter to chord bisects the chord & its intercepted arc

[then can use bisect to midpoint to congruent segs]

[then to congruent arcs]

6. diameter bisects a chord to the chord at midpoint of chord

7. [in 1 or s] 2 chords equidist. from center chords

8. 2 inscribed angles which intercept the same arc are

9. an angle inscribed in a semicircle is a right angle

10. quad inscribed in opp. angles are supplementary

11. parallel lines which intersect circle intercept arcs

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Angles

1. central = measure of intercepted arc

2. inscribed = 12 measure of intercepted arc

3. formed by tangent & chord has measure = 12 the intercepted arc

[notice this does not work for a secant and chord!]

4. formed by 2 chords which inside has measure = 12 sum of the 2

intercepted arcs

5. formed by 2 secants measure = 12

formed by 2 tangents difference of the

formed by 1 secant & 1 tangent intercepted arcs

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Segments

1. 2 chords inside

products of segments formed on each chord are =

2. 2 secant segs to

products of external seg & whole secant for each secant seg are =

3. secant & tangent seg to

product of ext. seg & whole secant = 2

tan seg