ganit pravinya past year papers

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Chapter 10: Circles

Section 1 - Basic Definitions:

1. In a plane, the set of all points that are a given distancefrom a given point is called a CIRCLE.

The given point is called the CENTER of the circle. The given distance is called the RADIUS of the circle. The symbol for circle is

2. A RADIUS of a circle is also defined to be a segmentfrom the centerof the circleto any point on the circle.

Radii of the same circle are congruent to each other.

3. A CHORD is a segment whose endpoints are both on the circle.

4. A DIAMETER of a circle is a chordwhich contains the centerof the circle.

The midpoint of a diameter of a circle is the center of the circle. The length of a diameter of a circle is twice the radius of the circle. Diameters of the same circle are congruent to each other.

5. Circles that have the same centerare called CONCENTRIC CIRCLES.

P

A

B

C PA = 1.7 cm.PB = 1.7 cm.

PC = 1.7 cm.

Point P is thecenter of the circle.

Since points A, B and C are all 1.7 cm. from

the center, theradius of the circle is 1.7 cm.

P

A

B

C

B

ASince points A and B are on the circle, then AB is a chord of the circle.

A P B

Point P is thecenter of the circle.

Since AB contains point P, then AB is adiameterof the circle.

Point P is the midpoint of AB

AB 2 PA

PA PB PC

P

AB

C Point P is the center of the circle with radius PA

Point P is the center of the circle with radius PB

Point P is the center of the circle with radius PC Since all three circles have thesame center, they

are said to beconcentric circles.


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mDC 40

DC is a minor arc

mGFE 360 40 320

GFE is a major arc

AB

ABmAB

6. A SECANT is a line that intersects a circle in exactly two points.7. A TANGENT is a line that intersects a circle in exactly one point.

1. The point of intersection is called the POINT OF TANGENCY.

8. A circle is measured in DEGREES. A circle contains 3609. An ARC of a circle, , consists of point A, point B and all points on the circle between them.

Arcs are also measured in degrees. The symbol, , represents the number of degrees in arc .

10.A SEMI-CIRCLE is an arc that is one-half of a circle. A semi-circle contains 180. The endpoints of a semi-circle are the endpoints of a diameter. A semi-circle is named with three letters. The first letter gives the starting point of the

semi- circle, the second letter names a point on the semi-circle and the third letternames the ending point of the semi-circle.

11.A MINOR ARC is an arc that is smaller than a semi-circle. A minor arc contains less than 180. A minor arc is named with only two letters, its starting point and its ending point.

12.A MAJOR ARC is an arc that is largerthan a semi-circle. A major arc contains more than 180. A major arc is named with three letters. The first letter gives the starting point of themajor arc, the second letter names a point on the major arc and the third letter names

the ending point of the major arc.

13.The MIDPOINT OF AN ARC is a point which separates the arc into two arcs whichcontain the same number of degrees.

G

T

F

H

C

FG is a secant that intersects the circle at points F and G

HC is atangent that intersects the circle at point TPoint T is called thepoint of tangency

40

BP

C

D

40

F EP

G

I HP

J

IH is a diameter of the circle

m IJH 180

IJH is a semi circle


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14. A CENTRAL ANGLE is an angle whose vertex is the centerof the circle andwhose sides are radii.

15.An INSCRIBED ANGLE is an angle whose vertex is on the circle and whosesides are chords.

16.The arc that the interior of a central angle or the interior of an inscribed angle cuts off iscalled an INTERCEPTED ARC.

Assignment: Section 1

1. In the diagram, point B is the centerof the circle.(a) ED is called a ____________________.(b)BC is called a ____________________.(c) DA is called a ____________________.(d)BD is called a _____________________.(e) Is BC BA ? ____________ Why?________________________________(f) Is BC DA ? ____________ Why?________________________________

2. Circles that have the same centerare said to be ____________________ circles.3. The midpoint of a diameterof a circle is the _______________.4. (a) In a plane, the set of all points a given distance from a given point is called a _________.

(b) The given point is called the _________

(c) The given distance is called the _________.

5. A segment whose endpoints are the center of a circle and apoint on the circleis called a _______________.

6. A segment whose endpoints are both on the circle is called a ________________7. A segment which has both endpoints on the circle but which alsopasses through

the centerof the circle is called a _________________.

8. A segmentthat intersects a circle in two points is called a ________________.9. A line that intersects a circle in two points is called a _________________.10.A line that intersects a circle in exactly one pointis called a __________________.

Point P is the center of the circle.

P is the vertex of APC so APC is a central angle.

APC intercepts AC

F is the vertex of GFH and point F is on the circle so

GFH is an inscribed angle.

GFH intercepts GH

B

C

D

A

E

P

C

A

H

F

G


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Thepoint of intersection is called the _____________________________.11.Arcs of circles are measured in _______________12.An arc that contains less than 180 is called a ___________________13.An arc that contains 180 is called a _____________________.14.An arc that contains more than 180 is called a ____________________15.A point which separates an arc into two arcs with equal measures is called a _____________16.The difference between a secantand a chordis that a secantis a ________________ but a

chord is a ______________.

17.A semi-circle is named with _______ letters.18.If an arc is named with two letters, than the arc must be a ______________ arc.19.A major arc is named with _________ letters.20. Point O is the center of the circle.

(a) FG is called a _________________

(b) FG is called a _________________

(c) OC is called a _________________

(d) ED is called a _________________

(e) AB is called a _________________

(f) HI is called a _________________

(g) Point G is called a _________________

(h) FG is called a _________________

(i) EAD is called a ________________

(j) CDE is called a ________________

21.In the diagram, point C is the center of the circle.(a) BCD is called a(an)_________________

(b) BAD is called a(an) _________________

(c) BCD intercepts what arc?_____________

(d) BAD intercepts what arc?_____________

(e) BC and CD are _______________

(f) AB and AD are ______________

EO

D

C

A

B

F G

H

I

CD

B

A


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22.In the diagram, point A is the center of the circle.(a) Name a central angle.

(b) Name an inscribed angle.

23.Refer to the diagram:Point R is the center of the circle.

(a)Name three radii:___________________(b)Name a diameter:___________________(c)Name a chord which is not a diameter:__________

24. If point O is the center of the circle, OA = 5x + 6 and OB = 2x + 24(a) Find x

(b) Find the radius of the circle(c) Find the diameter of the circle.

25.If point B is the center of the circle, BD = x + 4 and AC = 3x - 10,(a) Find x(b) Find the diameter of the circle(c) Find the radius of the circle.

26. If the radius of a circle is represented by "x", then which of the following would representthe diameter of the circle? (a) x (b) x (c) 2x

27. In the diagram, point O is the center of the circle.Tell whether each of the following is a major arc, a minor arc or a semi-circle.

28. In the diagram, point O is the center of the circle.(a) Name three chords.(b) Name three radii.(c) Name a tangent.(d) Name a secant.(e) Name a diameter.(f) Name a point of tangency.(g) Name two semi-circles

(h) Name five minor arcs.

A

D

C

F

E

B

R FA

W

TM

O

B

A

B

DA

C

(a)

(b) (f) (i)

(c) (g) (j)

(d)

(e) (h)ACB

AD ABC DCA

CAB CA DBA

CDB

BC ADBA

O B

C

D

CO

B

K

E

DH

G


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(i) Name two major arcs.29. In the diagram, point O is the center of the circle.

(a) AOC is called a(an)___________________

(b) ABD is called a(an) __________________

(c) COB is called a(an)___________________

(d) DEB is called a(an)___________________

(e) COB intercepts what arc?__________

(f) ABD intercepts what arc?_________

(g) DEB intercepts what arc?_________

30. (TF) Radii of the same circle are congruent to each other.31. (TF) A diameter of a circle is always a chord of the circle.32. (TF) A chord of a circle is always a diameter of the circle.33. (TF) The midpoint of any chord of a circle is the center of the circle.34. (TF) A chord of a circle intersects the circle in two points.35. (TF) A radius of a circle intersects the circle in two points.36. (TF) The midpoint of a radius of a circle is the center of the circle.37. (TF) Diameters of the same circle are congruent to each other.38. (TF) Every radius of a circle is also a chord of the circle.39. (TF) The midpoint of a diameter of a circle is the center of the circle.40. Which of the following is not a segment?

(a) a radius of a circle (b) a chord of a circle (c) the center of a circle

41. If the diameter of a circle is represented by "x" then which of the following wouldrepresent the radius of the circle? (a) x (b) x (c) 2x

42. Circles that have the same center are called ____________________ circles.43. If the diameter of a circle is 18, then the radius of the circle is ___________.44. (TF) A chord is a secant.45. (TF) If a line intersects a circle, then it must intersect the circle in two points.46. (TF) A secant has a midpoint.47. (TF) The midpoint of a tangent is the point of tangency.48. (TF) A minor arc is always named with two letters.49. Given the circle with center O. If AC = 5x - 1 and BD = 3x + 5,

(a) Find x(b) Find the diameter of the circle O

C

A

B

D

AO

B

C

DE


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(c) Find the radius of the circle.50. In the diagram, point O is the center of the circle.

If OA = 2x - 3 and AB = 3x - 1,

(a) Find x

(b) Find the diameter of the circle(c) Find the radius of the circle.

Section 2Measuring Central And Inscribed Angles

1. The measure of a central angle

is equal to the measure of its

intercepted arc.

2. The measure of an inscribed angle

is equal to one-half the measure of

its intercepted arc.

3. A triangle inscribed in a

semi-circle is a right triangle

and the diameter of the circle

is the hypotenuse of the triangle.

4. The opposite angles of a quadrilateralinscribed in a circle are supplementary.

For a quadrilateral to be inscribed in a circle,all four vertices of the quadrilateral must be

located on the circle.

Examples:

1. In the circle with center A, m CAB = 100.Find the number of degrees in arcs x and y.

O BA

A

B

C

m CAB mBC

B

C

A

1m ACB m AB

2

A C

B

D

ABC is an inscribed angle which

intercepts semi-circle ADC .

Therefore, m ABC 90

ABC is aright triangle

where hypotenuse AC is the diameter of the circle.

D

A

C

B

m A m C 180

m B m D 180

y

x

100 A

B

C

CAB is acentral angle.Since the measure of a central angle is equal to the measure of its

intercepted arc, if m CAB 100 then mBC 100 ,

so, x = 100

To find y, we know thatminor arc "x" plusmajor arc "y" must

add up to 360.

We subtract 360 - 100 = 260 so y = 260


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150

120

90 B

A

C

BAC is an inscribed angle which intercepts BC .Therefore, m BAC = mBC = (150) = 75

ABC is an inscribed angle which intercepts AC .

Therefore, m ABC = mAC = (120) = 60

ACB is an inscribed angle which intercepts AB .

Therefore, m ACB = mAB = (90) = 45

2. Given the circle with center P. If mAB = 80, find x and y.

3. In the circle with center P, mAC = 106. Find m ACB .

4. ABC is inscribed in the circle.The ratio of mAB mBC mAC: : = 3 : 5 : 4.

(a) Find mAB mBC and mAC, (b) Find each angle of the triangle.

80

yx

P

A B

PA and PB are radii of the circle,

therefore, they are congruent to each other.

This means APB is an isosceles triangle.

APB is acentral angle so m APB mAB Therefore, x = 80

To find thebase angles of the isosceles triangle, we subtract 80 from 180

and then split the difference between the two base angles.

180 - 80 = 100 and 100 2 50 . So, y = 50

106

xC

PB

A

CAB is asemi-circle so mAC mAB 180 . Therefore, mAB 74

ACB is an inscribed angle. Since an inscribed angle isequal to of its intercepted arc,

1 1x mAB 74 372 2

5x

4x

3x B

A

C

(a) Since the ratio of the three arcs is 3 : 5 : 4, we let mAB = 3x, mBC = 5x and mAC = 4x

Since the three arcs of the circle must add up to 360,We write the equation: 3x + 5x + 4x = 360

Combine like terms: 12x = 360

Divide by 12: x = 30

To find the number of degrees in the three arcs, we replace "x" with 30:

mAB = 3x = 3(30) = 90 and mBC = 5x = 5(30) = 150 and mAC = 4x = 4(30) = 120

(b) To find the three angles of the triangle, we notice that each angle of the triangle is aninscribed angle which is equal to the measure of its intercepted arc.


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D

A

C

BSinceopposite angles of an inscribed quadrilateral

are supplementary, m A m C 180

Therefore, m C = 180 - 75 = 105

5. Given the circle with center P. Find x and y.

6. Quadrilateral ABCD is inscribed in the circle. If m A = 75, find m C .


1. In each of the following, point P is the center of the circle. Find x and y:

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

P

BA x

y

270

P

C

AB

112x

y

P

BA

xy

128

P

BA

xy

310

P

BA

x

y

40

C

A

B100

110

x

y

C

A

B

100110

x

y

B

A

C P

y

118 x

3x

y

x

AP

B

CSince AB is adiameter, ACB is asemi-circle.

We write the equation: x + 3x = 180


Divide by 4: x = 45

CBA is an inscribed angle and therefore is equal to one-half of its intercepted arc.

So, y = mAC = (45) = 22.5

y x123P

B

A


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2. Quadrilateral ABCD is inscribed in the circle.If m B = 102, find m D .

3. In each of the following, point P is the center of the circle. Find x and y.(a) (b)

4. ABC is inscribed in the circle.The ratio of mAB mBC mAC: : = 5 : 4 : 11

(a) Find mAB mBC and mAC, .

(b) Find each angle of the triangle.

5. ABC is inscribed in the circle.The ratio of m A m B m C: : = 6 : 7 : 23.

(a) Find each angle of the triangle.

(b) Find mAB mBC and mAC,

6. Point E is the center of the circle.The diameter of the circle is 10 and BC = 8.

(a) Find m B .(b) Find AB

7. In each of the following, point P is the center of the circle. Find x and y:(a) (b) (c)

C

D

B

A

PBA

C

2xx

y

C

A

B

C

A

B

E CA

B

P

B

A

x

y52

P

BA

x

y

35

BC

A

P

x144

y

PBA

C

2x3x

y


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8. In each of the following, if given, point P is the center of the circle. Find x and y:(a) (b) (c)

9. ABC is inscribed in the circle.The ratio of m A m B m C: : = 7 : 5 : 3.

(a) Find each angle of the triangle.

(b) Find mAB mBC and mAC,

10.In each of the following, point P is the center of the circle. Find x and y:

11.Quadrilateral ABCD is inscribed in the circle.If m C = 138, find m A .

12.Point E is the center of the circle.The radius of the circle is 13 and AB = 10.(a) Find m B .(b) Find BC

13.ABC is inscribed in the circle.mAB= x + 15, mBC= 6x + 10 and mAC = 8x - 40

(a) Find mAB mBC and mAC,

(b) Find each angle of the triangle.

(c) Which of the following is true?

(i) ABC is an isosceles triangle

(ii) ABC is a scalene triangle

(iii) ABC is an equilateral triangle

CA

B

46210

x

y

P

BAxy

214

P

C

AB

x

y36

C

A

B

PBA

C

9x11x

yP

B

A

xy

218C

A

B

72

110x

y

C

D

B

A

E CA

B

CA

B


12/37


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7. To find the centerof any circle, find thepoint of intersection of theperpendicular bisectors ofany two chords of the circle.

Examples:

1. In the diagram, BC DE . mBC= 2x + 5 and mDE = 5x -10(a)Write the theorem that can be used to find x(b)Find x.

2. In the diagram, point P is the center of the circle. mAB= 3x and mDC = 2x + 13.(a)Why is BPA CPD ?(b)Write the theorem that can be used to find x(c)Find x.

3. In the diagram, point P is the center of the circle and PE AB .AM = 4x + 12 and MB = 6x - 10.

(a) PE is a __________ and AB is a __________(b)Write the theorem that can be used to find x(c)Find x.

T

L

PC

D

B

A

BC

D

E

P

B

A D

C

3x 2x + 13

L is the perpendicular bisector of AB

T is the perpendicular bisector of DC

Point P, the point of intersection of the

two perpendicular bisectors, is thecenter of the circle.

(a) Congruent chords intercept congruent minor arcs,

(b) We write the equation: 2x + 5 = 5x - 10

Subtract "2x" from both sides: 5 = 3x - 10

Add 10 to both sides: 15 = 3x

Divide by 3: 5 = x

(a) BPA and CPD are vertical angles so they are congruent.

(b) Congruent central angles intercept congruent arcs.

(c) We write the equation: 3x = 2x + 13Subtract "2x" from both sides: x = 13

A B

P

E

M

(b) A radius that is perpendicular to a chord bisects that chord.

(c) We write the equation: 4x + 12 = 6x - 10

Subtract "4x" from both sides: 12 = 2x - 10

Add 10 to both sides: 22 = 2x

Divide by 2: 11 = x


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4. In the diagram, point P is the center of the circle. PM = PN.CD = 5x + 2 and AB = x + 14.

(a)Write the theorem that can be used to find x(b)Find x.

5. The diameter of a circle with center P is 10. PC AB PD = 3.(a)Find the length of PA (b)Find the length of AD (c)Find the length of chord AB

6. In the diagram, DC AB mAD 20 and mDC 110. .(a)Write the theorem that can be used to find mBC (b)Find mBC (c)Find x.

BA

D C

20

110

x

BA

D C

20

110

x

20

P

B

A

C

DM

N

(a) If two chords are the same distance from the center of a circle,then they are congruent to each other.

(b) We write the equation: 5x + 2 = x + 14

Subtract "x" from both sides: 4x + 2 = 14

Subtract "2" from both sides: 4x = 12

Divide by 4: x = 3

53

A B

P

C

D

(a) Since the diameter of the circle is 10, then the radius of the circle is 5,therefore, PA = 5

(b) Since PC AB , APD is a right triangle.We may use thePythagorean Theorem to find the length of AD. However, in this case, we may use the 3 - 4 - 5 Pythagorean Triple and we

find that AD = 4.

(c) A radius that is perpendicular to a chord bisects the chord.

So, if AD = 4, then AB = 8.

(a) Parallel chords intercept congruent arcs between them(b) If mAD 20 then mBC 20, (c) Since a circle contains 360, we add mAD mDC and mBC, :

20 + 110 + 20 = 150

We then subtract 150 from 360: 360 - 150 = 210.

Therefore, x = 210

3A B

P

C

D


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7. In the diagram, point P is the center of the circle, AB CD and mCD 50 (a)Write the theorem that can be used to find mBD (b)Find mBD


1. Complete each of the following theorems by filling in the blank:(a) In a circle, congruent chords intercept ____________________________.(b) In a circle, if two chords are the same distance from the centerof the circle,

then they are ____________________.

(c) A radius which isperpendicularto a chord______________ the chord and___________ its minor arc.

(d) In a circle, if two chords areparallel, then the arcs between the two chords are___________

(e) If two central angles are congruent, then their intercepted arcs are ___________2. In the circle with center O, OA BC . BD = 7x - 3 and DC = 5x + 15,

(a) OA is a ___________, BC is a _____________(b) Write a theorem that can be used to find x(c) Find x.(d) Find BD(e) Find BC

3. In the diagram, point P is the center of the circle. APB DPC .mAB= 3x + 7 and mDC

x + 29.

(a) APB and DPC are both ___________angles.(b) Write a theorem that can be used to find x(c) Find x(d) Find mAB

(a)Parallel chords intercept congruent arcs between them.(b) If mAC then mBDx, x

ACB is asemi-circle and the three arcs, AC, CD, and DB

must add up to 180

Add the three arcs: x + 50 + x = 180

Combine like terms: 2x + 50 = 180

Subtract 50 from both sides: 2x = 130

Divide both sides by 2: x = 65

PA B

DC

50

x

PA B

DC

50

x x

O

CB D

A

P

D

C

BA


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B

CD

A

x

17

211

P

D

BA

Cx

12

P

D

BA

C

x

54B

CD

A

x

70

146

B

CD

A

x65

4. In each of the following AB CD . If point P is given, it is the center of the circle. Find x.(a) (b) (c)

5. In the diagram, AB CD . mAB= 3x + 4 and m CD= 5x16.(a) Write the theorem that can be used to find x.(b) Find x(c) Find mAB

6. Point O is the center of the circle and OC AB . mAC = 4x + 6 and mBC= 5x + 2,(a) Write the theorem that can be used to find x.(b) Find x(c) Find mAC (d) Find mAB

7. In the circle with center P, PD = 5 and PC AB .If the diameter of the circle is 26,

(a)Find the length of PA (b)Find the length of AD (c)Find the length of chord AB


9. In the diagram, AB CD . mAD = 7x + 12 and mBC= 14x - 16.(a) Write the theorem that can be used to find x

(b) Find x

B

C

D

A

O

BA

C

P

B

C

A D

5

B

CD

A

x

35

126

B

CD

A


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P

D

BA

C

30

x

B

CD

A

100

40

x

B

CD

A

x

80

120

B

CD

A

x

100

40P

D

BA

Cx

80

B

CD

A

x

170

60

10. The diameterof the circle with center P is 20and the length ofchord AB is 16.

PC AB

(a) Find the length of PA (b) Write a theorem that can be used to find AD(c) Find PD

11. In the circle with center O, OE = OF. AB = 2x - 3 and DC = 5x - 12,(a) Write a theorem that can be used to find x(b) Find x

(c) Find CD

12. In the circle with center O, OT RS .RD = 6x + 8 and DS = 10x - 36

(a) Write a theorem that can be used to find x.(b) Find x(c) Find RD(d) Find RS


(d) (e) (f)

14. In the diagram, point O is the center of the circle. mAB= 5x + 1 and mDC = 3x + 15.(a) AOB and DOC are not only central angles, they are also ______________ angles.

(b) Write the theorem that can be used to find x(c) Find x.

P

B

C

A D

B

O

DC

A

E

F

O

SR D

T

O

C

DA

B


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T

Q

R

P M

15.In the diagram, AB DC . mAB = 9x + 10 and mDC = 4x + 60.(a) Write the theorem that can be used to find x(b) Find x

16.The diameter of the circle with center T is 50. TR PQ and TM = 7(a) Find PT(b) Write a theorem that can be used to find PQ.(c) Find PQ.

17.In the diagram, point O is the center of the circle. OF = OE.AB = 6x - 5 and DC = 7x22.

(a) Write a theorem that can be used to find x.(b) Find x.

18.In the circle with center O, CD AB .mAD = 5x - 1 and mDB= 2x + 11(a) Write a theorem that can be used to find x

(b) Find x

19.In the circle whose center is point T, TR PQ If PQ = 30and TM = 8, find the diameter of the circle.

Section 4 - Tangent Theorems:

1. If a tangent is draw to a circle from an external point, the segment whose endpointsare the external pointand the point of tangency is called a TANGENT SEGMENT.

2. The two tangent segments drawn to a circle from an external point are congruentto each other.

B

C

D

A

O

B

A

CD

E

F

O

BA E

D

C

T

Q

R

P M

B

AAB is a tangent segment from external point A

to point of tangency B

B

P

A

PA PB


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3. The radius drawn to a tangent segment at the point of tangency is perpendicular

to the tangent segment.

4. The sum of the number of degrees in the angle formed by two tangents drawn to a circle

from an external point and the number of degrees in the minor arc intercepted by that angle

is 180.

Examples:

1. In the diagram, PB = 3x - 5 and PA = 2x + 3.

(a) Write a theorem that can be used to find x(b) Find x

2. Points A, B and C are points of tangency.(a) Find x

(b) Find y(c) Find PQ.

(a) In the diagram, PA and PB aretangent segments to

the circle from point P, so they are congruent.

O

A Ptangent

radius

OA AP

Q

R

P

m P mQR 180

B

P

A

BP Q

AC

7 3

x y

(a) The two tangent segments drawn to a circle from an externalpoint are congruent to each other

(b) SincePB = PAWe write the equation: 3x - 5 = 2x + 3

Subtract "2x" from both sides: x - 5 = 3

Add 5 to both sides: x = 8

(b) Likewise, QB and QCaretangent segments to the circlefrom point Q, so they are congruent.

Since QC = 3, then QB = 3 and so y = 3

(c) To find PQ, add x + y: 7 + 3 = 10, so PQ = 10


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3. Points P, Q and R are points of tangency. Find x.

4. In the diagram, m PBA = x + 4 and m P = 5x + 4. Find x.

5. Point O is the center of the circle and point A is a point of tangency.(a) Find x (b) Find y

6. Point O is the center of the circle. Find PO.

R

QP

A B

C

5

7

16

x

CQ and CP aretangent segments to the circle

from point C. Since CQ = 5, then CP = 5.

If CA = 16 and CP = 5,

we subtract 5 from 16 so that AP = 11

AP and AR aretangent segments to the circle frompoint A. Since AP = 11, then AR = 11

BQ and BR aretangent segments to the circle from

point B. Since BQ = 7, then BR = 7

To find x, we add AR + BR = 11 + 7 = 18.

Sincebase angles of an isosceles triangle are congruent,

if m PBA = x + 4, then m PAB = x + 4

Since the angles of any triangle add up to 180, m PAB m PBA m P 180

We write the equation: x + 4 + x + 4 + 5x + 4 = 180

Combine like terms: 7x + 12 = 180

Subtract 12 from both sides: 7x = 168

Divide by 7: x = 24

PA and PB are tangent segments to the circle

from point P so PA = PB.

Therefore, PAB is an isosceles triangle.

5x + 4

x + 4

A

B

P

O

A

P

x

25

y

O

A P

x5

12

(a) In the diagram, OA is a radius andPA is atangent segment. Therefore, OA PA

So, x = 90

(b) Sincethe acute angles of a right triangle arecomplementary, we subtract 25 from 90

and find that y = 65

In the diagram, OA is aradius and

PA is atangent segment. Therefore, OA PA

This means that OAP is a right triangle.

We use the 5 - 12 - 13 Pythagorean Triple

to find that x = 13


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7. Given two concentric circles. A chord ABof the larger circleis tangentto the smaller circle. If the radius of thelarger circle is 41 and the radius of the smaller circle is 9,

find the length of chord AB .

8. In the diagram, mBDA = 240. Find m C .


1. Points A, B and C are points of tangency.(a) Find x

(b) Find y

(c) Find TS

2. BA and BC are tangent segments to the circle from point B.

AB = 8x - 7 and BC = 2x + 35

(a) Write a theorem that can be used to find x(b) Find x(c) Find AB

A B

P

T

A B

P

T

To solve this problem, we will have todraw in radius PA .

PA = 41 and PT = 9.

Since PT is aradius and AB is atangent, we know that PT AB so PTA is aright triangle.

Since the radius of the larger circle is 41

and the radius of thesmaller circle is 9,

we use the 9 - 40 - 41 Pythagorean Triple to find that AT = 40.

Sincea radius that is perpendicular to a chord bisects the chord,

then PT bisects chord AB .Therefore, if AT = 40, then AB = 80.

419

A B

P

T

Since the number of degrees in a circle is 360,

we know that themajor arc BDA

plus theminor arc BA must add to 360

360 - 240 = 120 so mAB = 120

Since C is formed by twotangent segments from point C, we know that m C mAB 180

180 - 120 = 60 so m C 60

BT S

A C

yx

35

A

B

C

8x - 7

2x + 35

A

C

B

D

x240


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3. In each of the following, tangent segments are drawn to the circlefrom an external point. Find x.

(a) (b)

(c) (d)

4. In the diagram, points P, Q and R are points of tangency.CQ = 3, BQ = 4 and AB = 10. Find CA.

5. In the diagram, andBA BC are tangent segments.m B = 2x - 16 and m BAC = 3x + 14.

(a) Find x

(b) Find m BAC

(c) Find m ABC

6. In the circle with center O, point A is a point of tangency.OA is a __________ and PA is a __________________

If OA = 6 and PA = 8. Find PO.

7. CB and CA are tangent segments drawn

to the circle from point C.

If m C = 42, find mAB .

8. In the circle with center O, point A is a point of tangencyand m AOP = 50.

(a) Find x

(b) Find y

C

A

B

30

x

J

H

G

30

x

J

H

G

x80

R

P

T

x80

A

B

C

O

A

P

x6

8

A

C

B

x 42

O

A

P 50

x

y

R

QP

A B

C

3

4

x

10


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9. PQ and PR are tangent segments drawn to the circle from point P. If mQTR = 210,then find m P

10. Given two concentric circles. A chord of the larger circleis tangent to the smaller circle. If the radius of the

larger circle is 26 and the radius of the smaller circle is 24,find the length of the chord.

11.In the diagram, B, C and D are all points of tangency.AD = 12.

(a) Find x

(b) Find y

12. PA and PB are tangent segments to the circle from point P.

If PA = 7x - 1 and PB = 3x + 15,

(a) Find x

(b) Find PA

13. In the diagram, P, Q and R arepoints of tangency. CP = 2, PA = 5

and BC = 12. Find AB.

14.PQ is tangent to the circle whose center is R. If m Q = 38,(a) Find x

(b) Find y

15.Given two concentric circles. A chord of the larger circleis tangent to the smaller circle. The length of the chordis 30 and the radius of the smaller circle is 8.

Find the radius of the larger circle.

R

P

Q

T

x210

P

T BA

C

A

B

D12

x

y

A

P

B3x + 15

7x - 1

P Q

R

C

BA

12

x

5

2

R

P

Q

yx

38

A

RQP


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16.In the diagram, PQ and PR are tangent segmentsdrawn to the circle from point P.

If m P = 76, find mQTR .

17.In the diagram, andPA PB are tangent segments.m P = 3x - 5 and m PAB = 5x - 5.

(a) Find x

(b) Find m P

(c) Find m PAB 18. In each of the following, tangent segments are drawn to the circle from an external point. Find x.

(a) (b)

(c) (d)

19. In the diagram, tangent BA is drawn to the circle with center P.If PB = 25 and BA = 24,

(a) Find the radius of the circle

(b) Find the diameterof the circle

(c) Find the circumference of the circle

in terms of C=2r

(d) Find the area of the circle

in terms of 2Area=r

20. In the diagram, points E, B and D are points of tangency.AE = 4 and CD = 11. Find AC.

R

P

Q

T

x 76

A

P

B

AC

B

x

47

A

CB

x 28

A

C

B

x62

A

C

B

x72

P

AB 24

25

BA C

ED

411

x


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335

21.In the diagram, andPQ PR are tangent segments.m P = 5x and m PRQ = 4 x + 12.

(a) Find x

(b) Find m P

(c) Find m PRQ

22.Given two concentric circles. A chord of the larger circleis tangent to the smaller circle. If the radius of the

larger circle is 25 and the length of the chord is 48,

find the radius of the smaller circle.

23.PQ and PR are tangent segmentsdrawn to the circle from point P.

If mQTR = 196, find m P .

24.Points C, D and E are points of tangency.(a) Find x

(b) Find y

Section 5 - Other Angles Associated with Circles:

1. An angle formed by a line, segment or ray that is tangentto a circle and a chord drawn

at the point of tangency is measured by one-half its intercepted arc.

2. An angle formed by two chords intersecting within the interior of a circle is measured by

one-half the sum of its intercepted arc and the arc intercepted by its vertical angle.

Q P

R

A

R QP

R

P

Q

T

x196

BD

E

C

y

x

17

B

A

C

ABC is formed by tangent

BC and chordBA .

1m ABC m AB

2

E

CA

BD BEC is formed by chords

AB and CD intersecting at E.

1m BEC mBC m DA

2

Angle = (arc)

Angle = (arc + arc)


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B C

A

4x

5x

y

3. An angle formed by two secant segments intersecting at a point outside a circleis measured by one-half the difference of the larger intercepted arc and the smallerintercepted arc.

4. An angle formed by a tangent and a secant segment intersecting at a point outside a circleis measured by one-half the difference ofthe larger intercepted arc and the smaller

intercepted arc.

Examples:

1. Find x and m ABC.

ABC is formed by a tangent and a chord so it is measured by one-half of its intercepted arc.

Therefore, m ABC = m AB = (160) = 80

2. In the diagram, DE is tangent to the circle at point A. ABC is inscribed in the circle.

(a) Find m BAE (b) Find m CAD

BAE is formed by a tangent and a chord so it is measured by one-half of its intercepted arc.

A ED

B

C48

60

72

D

C

E

B

A

E is formed by secant

EA and secant EB .

1m E m AB mCD

2

Angle = (arcarc)

B

A E

C

E is formed by

tangent EA and secant EC

1m E m AC m AB

2

Angle = (arcarc)

Since the major arc and the minor arc must add to 360,

We write the equation: 5x + 4x = 360


Divide by 9: x = 40

Since x = 40, m AB = 160

First, we must find m AB :

BCA is an inscribed angle so it is measured

byone-half its intercepted arc.

If m BCA = 48, then m AB = 96

If m AB = 96, then m BAE = (96) = 48

Since CBA is an inscribed angle, m AC= 144.Therefore, m CAD = (144) = 72


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Since the arcs of a circle must add to 360:

We write the equation: 5x + x + 2x + 4x = 360



To find y, we will need mBC and m AD :

To find these arcs, we replace "x" with 30:

mBC = 2x = 2(30) = 60 and

m AD = 5x = 5(30) = 150

3. Find m BEC.

4. In the diagram, m BEC = 87 and mBC= 92. Find m AD

5. In the diagram, chords AB and CD intersect at point E. Find x and y.

C

D

A

B

E

100

120

110

x

C

B

A

D

E

8792

x

C

B

A

D

E y

x5x

2x

4x

C

B

A

D

E y

150

60

Since a circle contains 360, we find m AD :100 + 120 + 110 = 330 and 360 - 330 = 30

So, m AD = 30

Since BEC is formed by two chords, we use the formula:

Angle = (arc + arc)

1x mBC m AD2

1 1120 30 150 75

2 2

Since BEC is formed by two chords, we use the formula:

Angle = (arc + arc)

Substitute into the formula:1

87 92 x

2

Multiply both sides by 2:1

2 87 2 92 x2

Double the 87, cancel the : 174 = 92 + x

Subtract 92 from both sides: 82 = x

Since "y" is formed by two chords, we use the formula:

Angle = (arc + arc)1

y 60 1502

1y 210 105

2


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6. In the diagram, m AB = 84 and m DC= 14. Find m E

7. In the diagram, point P is the center of the circle. m AC= 4x, mCD = 6x and mBD = 2x.

(a) Find x (b) Find m E

To find m E, we need to find m AC and mBD . To find these arcs, we replace "x" with 15:

m AC = 4x = 4(15) = 60 and mBD = 2x = 2(15) = 30

8. In the diagram, EA is tangent to the circle at point A. m AC= 6x, mBC= 7x and m AB = 2x.

Find m E.

EA

C

B6x

7x

2x y

Since E is formed by two secants, we use the formula:

Angle = (arcarc)

1x 84 14

2

1x 70 352

E

CA

B

D

8414 x

Since P is the center of the circle, then AB

is a diameter and ACB is a semi-circle.

Therefore, m AC mCD mDB 180

We write the equation: 4x + 6x + 2x = 180

Combine like terms: 12x = 180Divide by 12: x = 15

ABP

E

C

D4x

6x

2x y

Since "y" is formed by two secants, we use the formula:

Angle = (arcarc)

1m E m AC mBD

2

1 1

y 60 30 30 152 2

A BPE

C

D60 30

Since the arcs of a circle must add up to 360,

We write the equation: 6x + 7x + 2x = 360



To find m E, we need to find m AC and m AB .

To find these arcs, we replace "x" with 24: m AC = 6x = 6(24) = 144

and m AB = 2x = 2(24) = 48


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9. In the diagram, EA is tangent to the circle at point A. If m E = 24 and m AC= 112,

find m AB .


1. In each of the following, AB is a chord and BC is a tangent.(a) (b) (c)

m AB _______

mBDA _______

m ABC 65

(d) (e)

Since E is formed by a tangent and a secant, we use the formula:

Angle = (arcarc)

1m E m AC m AB

2

1 1y 144 48 96 48

2 2

EA

C

B

y48

144

EA

C

B

24

112

x

Since E is formed by a tangent and a secant,

we use the formula:

Angle = (arcarc)

We write the equation:1

24 112 x2

Multiply both sides by 2:1

2 24 2 112 x2

Double the 24 and cancel the : 48 = 112 - x

Subtract 112 from both sides: - 64 = - x(48 - 112 = - 64)

Divide both sides by - 1: 64 = x

x110

B

A

C B

A

C

D

x = _________

m ABC = ________

9x3x

B

A

Cx = _________

m ABC = ________

7x

3x

B

A

C

D

210

B

A

C

m ABC ________

m AB 110

m AB ______

m ABC ______

m ADB 210


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32/37


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14. Find "x" and "y" in each of the following: If point P is given, P is the center of the circle.)(a) (b)

(c) (d)

15. Find "x" in each of the following:(a) (b)

Section 6Product Relationships Between Segments Associated with Circles:

1. If two chords intersect within a circle, then the product of the segments of one chordis equal to the product of the segments of the other chord.

Chords andAB CD intersect at point E.

EA EB ED EC

2. If two secant segments are drawn to a circle from an external point, the product of oneentire secant segment and its external segment is equal to the product of the other entire

secant segment and its external segment.

Secant segments andEA ED are drawn from point E.

EA EB ED EC

A E

C

B9x

7x

2x y

BA

P

C

E

D

y

7x

5x

3x

C

E

D

B

A

y

10x

8x

13x

5x

C

D

AB

E

P

y

7x

2x

5x

C

E

D

B

A

95250 x

C

D

A

BE162

228

X

B

A

D

C

E

CD

E

AB


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3. If a secant segment and a tangent segment are drawn to a circle from an external point, the

product of the entire secant segment and its external segment is equal to the square

of the tangent segment.

Examples:

1. Chords andWX YZ intersect at point P.WP = x + 3, PX = 3, PZ = x - 4 and PY = 4. Find x

2. Tangent segmentBC and secant segment BA are drawn to the circle from point B.If AB = 8 and AD = 6, find BC.

3. Tangent segment AD and secant segment AB are drawn to the circle from point A.If AD = 3 and AB = 9, find AC.

A E

B

C

XW

Z

Y

Px + 3 3

x - 4

4

C B

A

D

x

6 8

D

AB C x

3

9

Secant segment EB and tangent segment

EA are drawn to the circle from point E.2

EB EC EA

We write the formula: PW PX PZ PY

Substitute the given values: x 3 3 x 4 4

Distribute the "3" and the "4": 3x + 9 = 4x - 16

Subtract "3x" from both sides: 9 = x - 16

Add 16 to both sides: 25 = x

Since we wish to find BC, we let BC = x

We write the formula:2

BA BD BC

To find BD, we subtract 6 from 8: BD = 2

Substitute:2

8 2 x

Multiply:216 x

Take the square root of 16: 4 = x

Since we wish to find AC, we let AC = x.

We write the formula:2

AB AC AD

Substitute: 29 x 3

Square the 3: 9x = 9

Divide by 9: x = 1


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4. Secant segments andCA CE are drawn from point C.If AB = 14, BC = 4, and EC = 24, find CD.


1. Chords andAB CD intersect at point E.

If AE= 9, EB = 4 and EC = 3, find ED.

2. Chords andAC DE intersect at point B. If AB = 8, BC = 2

and BD = 4, find BE.

3. Chords andAB CD intersect at point E.

If AE = 7, EB = x + 2, CE = 3 and

ED = 2x + 9, find x.

3. Tangent segment CD and secant segment CA are drawn frompoint C. If BC = 4 and AC = 9, find CD.

4. Tangent segment DB and secant segment DA are drawn from point D.

If CD = 4 and AC = 21, find BD.

5. Tangent segment BA and secant segment BD are drawn from point B.If BA = 12 and BD = 36, find BC.

6. Tangent segment BA and secant segment BD are drawn from point B. If AB = 8 and BC = 2,

(a) Find BD

(b) Find DC

DE

C

AB14

4

24

x

CD

A

B

E

A

E

D

C B

D

B

A

C

E

D

A

C

B

B

ADC

A

D

B

C

A

D

B

C

Since we wish to find CD, we let CD = x

We write the formula: CA CB CE CD

Since AB = 14 and BC = 4, then CA = 18

Substitute: 18 4 24 x

Multiply: 72 = 24x

Divide by 24: 3 = x


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8. Secant segments andAD AE are drawn from point A.

If AD = 34, AB = 2 and AC = 4, find AE.

9. Secant segments AB and ACare drawn from point A. If AB = 16,

AD = 3 and AC = 12, find AE.

10.Secant segments andBC BA are drawn from point B.If CD = 30, DB = 2 and AB = 16, find BE.

11.Secant segments andCA CE are drawn from point C. If BC = 4,AC = x + 3, DC = 5 and CE = x - 2,

find x.

12. Secant segments AB and AD are drawn from point A.

If AC = 3, AB = 2x - 4, AE = 2 and AD = 2x + 4, find x.

13. Chords andAC BD intersect at point E.

If AE = 14, EC = 3 and DE = 7, find EB.

14. Tangent segment CD and secant segment CA are drawn frompoint C. If AC = 4 and BC = 1, find CD.

15. Secant segments andCA CE aredrawn from point C. If BC = 7, AC = 3x - 7,

CD = 8 and CE = 2x + 7, find x.

16. Tangent segment CD and secant segment CA are drawnfrom point C. If BC = 3 and AB = 24, find DC.

17. Chords andWX ZY intersect at point P.

If WP = 3, PX = x + 4,PY = 5 and PZ = x - 2, find x.

A

C

D

E

B

A

E

B

C

D

B

E

C

A

D

CD

AB

E

AE

BC

D

B

C

A

D

E

D

ACB

CD

AB

E

D

A

C

B

X

Y

Z

W P


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18. Tangent segment BA and secant segment BD aredrawn from point B. If BA = 8 and BC = 4, find BD.

19. Tangent segments andCA CE are drawn from point C.

If CB = 3, AB = 22,CD = 5, find EC

20. Tangent segment AB and secant segment AD are drawnfrom point A. If AB = 12 and AD = 18,

(a) Find AC(b) Find DC

21. Chords andAC BD intersect at point E.

If AE = 36, EC = 2, and ED = 6, find EB.

22. Secant segments andAD AE are drawn from point A.

If AC = 2, DC = 25 and AB = 3(a) Find AE

(b) Find EB

23. Tangent segment PA and secant segment PCare drawnfrom point P. If PA = 10 and PC = 25, find PB

24. Chord DB bisects chord AC.If PB = 4 and PD = 16, find PA

25. Secant segments andAB AD are drawn from point A.If AC = 3, AB = x + 4, AE = 2 and AD = 4x + 1, find x.

26. Chords AB and CD intersect at point E.If AE = 4, EB = x - 5, CE = 3 and ED = x + 2, find x

A

D

B

C

C

D

AB

E

B

D AC

C

D

B

A

E

AB

DC

E

A

P

C

B

B

CD

A

Px

x

B

AD E

C

D

BC

AE

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