eleanor roosevelt high school chin-sung lin. mr. chin-sung lin erhs math geometry

Geometry of The Circle

Eleanor Roosevelt High School Chin-Sung Lin

Geometry Chapter 13

Arcs, Angles, and Chords

Mr. Chin-Sung LinERHS Math Geometry

CircleA circle is the set of all points in a plane that are

equidistant from a fixed point of the plane called the center of the circle

Circles are named by their center (e.g., Circle C)

Symbol: O

CCircle


CenterIt is the center of the circle and the distance from this

point to any other point on the circumference is the same

CCircle


Center

RadiusA radius is the line segment connecting (sometimes

referred to as the “distance between”) the center and the circle itself

CCircle


Center

rA

Radius

CircumferenceA circumference is the distance around a circle

It is also the perimeter of the circle, and is equal to 2 times the length of radius (2r)

CCircle


Center

rA

Radius

Circumference

ChordA chord is a line segment with endpoints on the

circle

CCircle


B

A

Chord

DiameterA diameter of a circle is a chord that has the center of

the circle as one of its points

CCircle


B ADiameter

ArcAn arc is a part of the circumference of a circle

(e.g., arc AB)

CCircle


B

A

Arc

Central AngleA central angle is an angle in a circle with vertex at

the center of the circle

(e.g., ACB)

CCircle


B

A

Arc

Major ArcGiven two points on a circle, the major arc is the

longest arc linking them

(e.g., arc ADB, mACB > 180)

CCircle


B

A

Major Arc D

Minor ArcGiven two points on a circle, the minor arc is the

shortest arc linking them

(e.g., arc AB, mACB < 180)

CCircle


B

A

Minor Arc

SemicircleHalf a circle. If the endpoints of an arc are the endpoints

of a diameter, then the arc is a semicircle

(e.g., arc ADB, mACB = 180)

CCircle


B A

SemicircleD

Adjacent ArcsAdjacent arcs are non-overlapping arcs with the same

radius and center, sharing a common endpoint

(e.g., arc AB and AD)

CCircle


B

A

Adjacent ArcsD

Intercepted ArcIntercepted Arc is the part of a circle that lies between

two lines that intersect it

(e.g., arc AB and XY)

CCircle


B

A

Intercepted Arcs

X

Y

Arc LengthAn arc length is the distance along the curved line

making up the arc

CCircle


B

A

Arc Length

Degree Measure of an ArcThe degree measure of an arc is equal to the measure

of the central angle that intercepts the arc

(e.g., m AB = mACB)

CCircle


B

A Measure of Central Angle

= Measure of Intercepted

Arc

Measure of a Minor ArcThe measure of minor arc is the degree measure of

central angle of the intercepted arc

(e.g., m AB = mACB)


CCircle

B

ADegree Measure of a Minor Arc

Measure of a Major ArcThe measure of major arc is 360 minus the degree

measure of the minor arc

(e.g., m ADB = 360 – mACB)


C Circle

B

ADegree Measure of a Major Arc

D

Congruent CirclesCongruent circles are circles that have congruent radii

(e.g., O ≅ O’)


O

Circle

A

Congruent Circles

O’

Circle

B

Congruent ArcsCongruent arcs are arcs that have the same degree

measure and are in the same circle or in congruent circles (e.g., AB ≅ CD ≅ XY)


O

Circle

ACongruen

t ArcsO’

Circle

X

BY

C

D

Concentric CirclesConcentric Circles are two circles in the same plane

with the same center but different radii


OConcentric Circles

A

X

Theorems


Congruent Radii In the same or congruent circles all radii are congruent

If C O, r, s and t are radii,

then r = s = t


Cr O

s

Congruent Radii

t

Congruent ArcsIn the same or in congruent circles, if two central

angles are congruent, then the arcs they intercept are congruent

If central angles ACB XOY,

then the intercepted arcs

AB XY


CB

A

OY

X

Congruent Central Angles = Congruent Arcs

Congruent Central Angles In the same or in congruent circles, if two arcs are

congruent, then their central angles are congruent

If the arcs AB XY,

then their central angles

ACB XOY


CB

A

OY

X

Congruent Arcs = Congruent Central Angles

Congruent Arcs & Central Angles

In the same or in congruent circles, two arcs are congruent if and only if their central angles are congruent

The arcs AB XY,

if and only if their central

angles ACB XOY


CB

A

OY

X

Congruent Arcs = Congruent Central Angles

Arc Addition PostulateIf AB and BC are two adjacent arcs of the same circle ,

then AB + BC = ABC and mAB + mBC = mABC


OCircle

A

B

C

Congruent ChordsIn the same or in congruent circles, if two central

angles are congruent, then the chords are congruent

If central angles ACB XOY,

then the chords AB XY


CB

A

OY

X

Congruent Central Angles = Congruent Chords

Congruent Central Angles In the same or in congruent circles, if two chords are

congruent, then their central angles are congruent

If the chords AB XY,

then their central angles ACB XOY


Congruent Chords = Congruent Central Angles

CB

A

OY

X

Congruent Chords & Central Angles

In the same or in congruent circles, two chords are congruent if and only if their central angles are congruent

The chords AB XY if and only if

their central angles

ACB XOY


Congruent Chords = Congruent Central Angles

CB

A

OY

X

Congruent ChordsIn the same or in congruent circles, if two arcs are

congruent, then the chords are congruent

If arcs AB XY,

then the chords AB XY


CB

A

OY

X

Congruent Arcs= Congruent Chords

Congruent ArcsIn the same or in congruent circles, if two chords are

congruent, then their arcs are congruent

If the chords AB XY,

then their arcs AB XY


Congruent Chords = Congruent Arcs

CB

A

OY

X

Congruent Arcs & Chords In the same or in congruent circles, two chords are

congruent if and only if the arcs are congruent

Arcs AB XY if and only if the chords AB XY


CB

A

OY

X

Congruent Arcs= Congruent Chords

Congruent Semicircles Postulate

The diameter of a circle divides the circle into two congruent arcs (semicircles)

If AB is a diameter of circle C, then APB AQB


C

Diameter AB

Q

P

Exercise 1 Circle C has central angle ACB = 60o, what’s the measure of

the arc ADB?

CB

AD


Exercise 2 Circle C has central angle ACB = 60o, DCE = 60o, and BCD =

170o, what’s the measure of the arc AD and BE?

CB

AD

E


Exercise 3 Circle C has diameter BD and EF. If central angle ACF = 90o,

DCE = 50o, what’s the measure of the arc DF, AE and BE?

CB

AD

E

F


Exercise 4The length of the diameter of circle C is 26 cm. The chord AB is

5 cm away from the center C. What is the length of AB?

26

5

C A

B

Y

X


Exercise 5 The length of the chord AB of circle C is 10. The circumference

of circle C is 20 . What’s the measure of arc AB?

CB

A


Exercise 6If two concentric circles have radii 10 and 6 respectively, what’s

the total area of the blue regions?

C10

6


Theorem of Chords


Chord Bisecting TheoremIf a diameter is perpendicular to a chord, then it bisects

the chord and its major and minor arcs

Given: Diameter CD ABProve:

1) CD bisects AB2) CD bisects AB and ACB

Circle

O

A B

C

D

M


Chord Bisecting TheoremIf a diameter is perpendicular to a chord, then it bisects

the chord and its major and minor arcs

Given: Diameter CD ABProve:

1) CD bisects AB2) CD bisects AB and ACB

Circle

O

A B

C

D

M


1 2

Secants, Tangents, and Inscribed Angles


SecantA secant is a segment or line which passes through a

circle, intersecting at two points

C

Secant

AB

D


TangentA tangent is a line in the plane of a circle that intersects the

circle in exactly one point (called the point of tangency)

C Tangent

Point of TangentB

A

D


Degrees/Radians of a CircleThere are 360 degrees in a circle or 2 radians in a circle

Thus 2 radians equals 360 degrees

360o or 2C A


Inscribed AngleAn inscribed angle is an angle that has its vertex and its sides

contained in the chords of the circle

(e.g., ADB)

CB

AD


Inscribed Angle

Inscribed PolygonAn inscribed polygon is a polygon whose vertices are on the

circle

Inscribed Polygon

CX

WZ

Y


Circumscribed PolygonCircumscribed polygon is a polygon whose sides are tangent

to a circle

Circumscribed Polygon

X

WZ

C

Y


Theorems of Inscribed Angles


Inscribed Angle TheoremThe measure of an inscribed angle is equal to one-half

the measure of its intercepted arc

Given: Inscribed angle ACBProve: mACB = (1/2) m AB

Circle

O

A B

C





Proof: (Case 1)Inscribed angles where one chord is a diameter

1

O

A B

C

2

3





Proof: (Case 2)Inscribed angles with the center of the circle in their interior

Circle

O

A B

C

1 2

3 4





Proof: (Case 3)Inscribed angles with the center of the circle

in their exterior

Circle

O

A B

C

D 123

45

6


Inscribed Angle TheoremGiven: Inscribed angle ACBProve: mACB = (1/2) m AB

m1 = m3 - m2m4 = m6 - m5m3 = 2 m6m2 = 2 m5 m3 - m2 = 2 (m6 - m5) = 2 m4m1 = 2 m4m4 = (1/2) m1 mACB = (1/2) m AB

Circle

D

O

A B

C

12

3

45

6


Congruent Inscribed Angle Theorem

In the same or in congruent circles, if two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent

Given: Inscribed angle ACB and ADB

Prove: ACB ADB


Circle

O

A B

C

D

Right Inscribed Angle TheoremAn angle inscribed in a semi-circle is a right angle

Given: Inscribed angle ACB and AB is a diameter

Prove: mACB = 90o

Circle

O

A

B

C


Right Inscribed Angle TheoremAn angle inscribed in a semi-circle is a right angle

Given: Inscribed angle ACB and AB is a diameter

Prove: mACB = 90o

Circle

O

A

B

C

180o


Supplementary Inscribed Angle Theorem

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary

Given: ABCD is an inscribed quadrilateral of circle O

Prove: mB + mD= 180

Circle

O

A

B

C

D


Parallel Chords and Arcs TheoremIn a circle, parallel chords intercept congruent arcs

between them

Given: AB || CDProve: AC BD

Circle

O

A B

C D


Exercises


ExerciseC has an inscribed quadrilateral ABCD where A = 70o and B

= 80o. What’s the measures of C and D?


70o

O

A

B

C

D

80o

ExerciseC has an inscribed quadrilateral ABCD where A = 70o and B

= 80o. What’s the measures of C and D?


70o

O

A

B

C

D

80o

100o

110o

Exercise

Mr. Chin-Sung Lin

C has an inscribed angle ADB = 30o, DB is the diameter. DEA =?

CB

A

D

30o

E

Exercise

Mr. Chin-Sung Lin

C has an inscribed angle ADB = 30o, DB is the diameter. DEA =?

CB

A

D

30o

E

60o

60o

Theorems of Tangents


Unique Tangent Postulate

At a given point on a circle, there is one and only one tangent to the circle

Given P is on the circle OThere is only one tangent APto circle O

Circle

O

A

P

Tangent


Perpendicular-Tangent Theorem

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of contact

Given: AB is a tangent to OP is the point of tangency

Prove: AB OP

Circle

O

A

P

B


Perpendicular-Tangent TheoremGiven: AB is a tangent to O

P is the point of tangencyProve: AB OP (Indirect Proof)

1. Suppose OP is NOT perpendicular to AB

2. Draw a point D on AB, OD AB

3. Draw point E on AB, PD = DE and

E is on different side of D

4. ODP = ODE = 90°

5. OD = OD (Reflexive)

Circle

O

A

P

BD

E


Perpendicular-Tangent TheoremGiven: AB is a tangent to O

P is the point of tangencyProve: AB OP (Indirect Proof)

6. ODP ODE (SAS)

7. OP = OE (CPCTC)

8. E is on O (by 7)

9. AB intersects the circle at two

different points, so AB is not

a tangent (contradicts to the given)

10. AB OP (the opposite of the assumption is true)

Circle

O

A

P

BD

E


If a line is perpendicular to a radius at its outer endpoint, then it is a tangent to the circle

Given: OP is a radius of O andAB OP at P

Prove: AB is a tangent to O

Circle

OA

P

Converse of Perpendicular Tangent Theorem

B


Converse of Perpendicular Tangent Theorem

Given: OP is a radius of O andAB OP at P

Prove: AB is a tangent to O

1. Let D be any point on AB other than P

2. OP AB (Given)

3. OD > OP (Hypotenuse is longer)

4. D is not on O (Def. of circle)

5. AB is a tangent to O (Def. of circle)

Circle

O

A

P

BD


Perpendicular-Tangent Theorem

If a line is tangent to a circle if and only if it is perpendicular to the radius drawn to the point of contact

Circle

O

A

P

B


Common Tangents

A common tangent is a line that is tangent to each of two circles

O


O’

A

B

Common Internal Tangent

O O’

A B

Common External Tangent

Common Tangents

Two circles can have four, three, two, one, or no common tangents


01234

Circles Tangent Internally/Externally

Two circles are said to be tangent to each other if they are tangent to the same line at the same point


Tangent InternallyTangent Externally

Tangent Segments

A tangent segment is a segment of a tangent line, one of whose endpoints is the point of tangency

PQ and PR are tangent segments of the tangents PQ and PR to circle O from P.

Circle

O

P

R

TangentSegments


Q

Congruent Tangents Theorem

If two tangents are drawn to a circle from the same external point, then these tangent segments are congruent

Given: AP and AQ are tangents to O, P and Q are points of tangency

Prove: AP AQ

O

A

P

Q


Congruent Tangents Theorem

If two tangents are drawn to a circle from the same external point, then these tangent segments are congruent

Given: AP and AQ are tangents to O, P and Q are points of tangency

Prove: AP AQ

(HL Postulate)

O

A

P

Q


Angles formed by Tangents Theorem

If two tangents are drawn to a circle from an external point, then the line segment from the center of the circle to the external point bisects the angle formed by the tangents

Given: AP and AQ are tangents

to O, P and Q are points of tangency

Prove: AO bisects PAQ

O

A

P

Q


Angles formed by Tangents TheoremIf two tangents are drawn to a circle from an external point,

then the line segment from the center of the circle to the external point bisects the angle whose vertex is the center of the circle and whose rays are the two radii drawn to the points of tangency.

Given: AP and AQ are tangents

to O, P and Q are points of tangency

Prove: AO bisects POQ

O

A

P

Q


Exercises


ExerciseCircles O and O’ with a common internal tangent, AB, tangent

to circle O at A and circle O’ at B, and C the intersection of OO’ and AB

(a) Prove AC/BC = OC/O’C

(b) Prove AC/BC = OA/O’B(c) If AC = 8, AB = 12, and OA = 9

find O’B

O


O’

A

B

C

ExerciseCircles O and O’ with a common internal tangent, AB, tangent

to circle O at A and circle O’ at B, and C the intersection of OO’ and AB

(a) Prove AC/BC = OC/O’C

(b) Prove AC/BC = OA/O’B(c) If AC = 8, AB = 12, and OA = 9

find O’B

(c) O’B = 9/2

O


O’

A

B

C

ExerciseC has a tangent AB. If AB = 8, and AC = 12. (a) What is exact length

of the radius of the circle? (a) Find the length of the radius of the circle to the nearest tenth

C

B

A

12D

8


ExerciseC has a tangent AB. If AB = 8, and AC = 12. (a) What is exact length

of the radius of the circle? (a) Find the length of the radius of the circle to the nearest tenth

(a) 4√ 5

(b) 8.9C

B

A

12D

8


ExerciseC has a tangent AB and a secant AE. If the diameter of the circle is

10 and AD = 8. AB = ?

C

B

A

8D10

E


ExerciseC has a tangent AB and a secant AE. If the diameter of the circle is

10 and AD = 8. AB = ?

C

A

8D

5

E5


B

ExerciseFind the perimeter of the quadrilateral WXYZ

8

Circumscribed Polygon &

Inscribed Circle

X

WZ

C

Y


A

B

C

D

5

4

ExerciseFind the perimeter of the quadrilateral WXYZ

Perimeter: 34 8

Circumscribed Polygon &

Inscribed Circle

X

WZ

C

Y


A

B

C

D

5

4

Angle Measurement Theorems



Measure an angle formed by

A tangent and a chord

Two tangents

Two secants

A tangent and a secant

Two chords


An Angle Formed by A Tangent and A Chord


Tangent-Chord Angle TheoremThe measure of an angle formed by a tangent and a

chord equals one-half the measure of its intercepted arc

Given: CD is a tangent to O, B is the point of tangency, and AB is a chord

Prove:1) mABC = (1/2) m AB2) mABD = (1/2) m AEB

O

A

BC D

E


Tangent-Chord Angle TheoremGiven: CD is a tangent to O, B is the point of tangency, and

AB is a chord

Prove:1) mABC = (1/2) m AB2) mABD = (1/2) m AEB 1 O

A

BC D

2

E


Tangent-Chord Angle Theorem

1. Draw OA and OB, form 1 and 22. OB CD3. mABC + m2 = 904. OA = OB 5. m1 = m2 6. m1 + m2 + mAOB = 180 7. 2m2 + mAOB = 180 8. m2 + (1/2) mAOB = 90 9. m2 + (1/2) mAOB = mABC + m210. (1/2) mAOB = mABC 11. mABC = (1/2) m AB12. 180 - mABC = (1/2) (360 - m AB)13. mABD = (1/2) m AEB


1 O

A

BC D

2

E

Tangent-Chord Angle ExampleIf CD is a tangent to O, B is the point of tangency, and ABE is an

inscribed triangle

what are the measures of ABC, EBD, AB and EAB ?

O

A

BC D

E70o

80o


Tangent-Chord Angle ExampleIf CD is a tangent to O, B is the point of tangency, and ABE is an

inscribed triangle

what are the measures of ABC, EBD, AB and EAB ?

O

A

BC D

E70o

80o

70o 80o

140o160o

60o


Angles Formed by Two Tangents, Two Secants

and A Secant A Tangent


Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Theorems

The measure of an angle formed by two tangents, by a tangent and a secant, or by two secants equals one-half the difference of the measure of their intercepted arcs

O

B

A

CE

OB

A

C

D

OB

A

C

D



Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC)

E

OB

A

C

D

OB

A

C

D

O

B

A

CE

E


Tangent-Tangent Angle Theorem (1)Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC)

1. Draw BC, form 1 and 2

2. AB = AC

3. m1 = m2 = (1/2) m BC

4. mA + m1 + m2 = 180

5. mA + m BC = 180

6. m BC + m BEC = 360

7. (1/2) m BC + (1/2) m BEC = 180

8. mA + m BC = (1/2) m BC + (1/2) m BEC

9. mA = (-1/2) m BC + (1/2) m BEC

10. mA = (1/2) (m BEC - m BC )

O

B

A

CE

1

2



1. Draw OB and OC, form 1 and 22. OB AB, OC AC3. mA + m1 + m2 + mBOC = 3604. mA + 90 + 90 + mBOC = 3605. mA + mBOC = 180 6. mA + m BC = 1807. m BC + m BEC = 360 8. (1/2) m BC + (1/2) m BEC = 180 9. mA + m BC = (1/2) m BC + (1/2) m BEC 10. mA = (-1/2) m BC + (1/2) m BEC 11. mA = (1/2) (m BEC - m BC )

O

B

A

CE

1

2



1. Draw BC, form 2

2. Extend AC, form 1

3. m2 = (1/2) m BC

4. m1 = (1/2) m BEC

5. mA = m1 - m2

6. mA = (1/2) m BEC - (1/2) m BC

7. mA = (1/2) (m BEC - m BC )

O

B

A

CE1

2


D


Given: O with secants AD and AE Prove: mA = (1/2) (m DE - m BC)

E

OB

A

C

D

OB

A

C

D

O

B

A

CE

E


Secant-Secant Angle Theorem (1)Given: O with secants AB and AC

Prove: mA = (1/2) (m DE - m BC)

1. Draw DC

2. m2 = mA + m1

3. mA = m2 - m1

4. m2 = (1/2) m DE, m1 = (1/2) m BC

5. mA = (1/2) (m DE - m BC)

A

CE

B

O

D

1

2


Secant-Secant Angle Theorem (2-1)Given: O with secants AB and AC Prove: mA = (1/2) (m DE - m BC)

1. Draw OB, OC, OD and OE

2. OB = OC = OD = OE

3. m3 = m4, m7 = m8

4. m5 = 180 - 2 m3, m9 = 180 - 2 m7

5. m3 = mBOA + mBAO

6. m7 = mCOA + mCAO

7. m5 + m9 = 180 - 2 m3 + 180 - 2 m7

= 360 - 2(m3 + m7)

A

CE

1

2

B

O

D

34

5

78

9


Secant-Secant Angle Theorem (2-2)Given: O with secants AD and AE Prove: mA = (1/2) (m DE - m BC)

8. m5 + m9 = 360 - 2(mBOA + mBAO + mCOA + mCAO) = 360 - 2(mBOC + mA)

9. m5 + m9 + 2(mBOC + mA) = 360

10. m5 + m9 + mBOC + mDOE = 360

11. mBOC - mDOE + 2mA = 0

12. 2 mA = mDOE - mBOC

13. mA = (1/2) (mDOE- mBOC)

14. mA = (1/2) (m DE- m BC)

A

CE

1

2

B

O

D

34

5

78

9



Given: O with a secant AD and a tangent AC Prove: mA = (1/2) (m DEC - m BC)

E

OB

A

C

D

OB

A

C

D

O

B

A

CE

E


Secant-Tangent Angle Theorem

Given: O with a secant AD and a tangent AC

Prove: mA = (1/2) (m DEC - m BC)

1. Draw BC

2. m2 = mA + m1

3. mA = m2 - m1

4. m2 = (1/2) m DEC, m1 = (1/2) m BC

5. mA = (1/2) (m DEC - m BC)

A

CE

1

B

O

D

2



The measure of an angle formed by two tangents, by a tangent and a secant, or by two secants equals one-half the difference of the measure of their intercepted arcs

O

B

A

CE

OB

A

C

D

OB

A

C

D


Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Example 1

If O with tangents AD, AC, secants GB and GD, calculate m BC, and mG

O

B

A

C E

G

D40o

50o

F



If O with tangents AD, AC, secants GB and GD, calculate m BC, and mG

O

B

A

C E

G

D40o

50o

F

65o

65o

130o

25o



If O with a tangent AB, secants AD, GB and GD, calculate m BD, and m BF

O

B

A

C E

H

G

D

F

30o

40o

20o



If O with a tangent AB, secants AD, GB and GD, calculate m BD, and m BF

O

B

A

C E

H

G

D

F

30o

40o

20o

100o

60o


An Angle Formed by Two Chords


Chord-Chord Angle TheoremThe measure of an angle formed by two chords intersecting

inside a circle equals one-half the sum of the measures of its intercepted arcs

Given: O with chords AB and CD Prove: mAMC = mBMD

= (1/2) (m AC + m BD)


O

A

BC

D

M

Chord-Chord Angle TheoremGiven: O with chords AB and CD Prove: mAMC = mBMD = (1/2) (m AC + m BD)

1. Draw BC

2. mAMC = mBMD 3. mAMC = m1 + m24. m1 = (1/2) m AC5. m2 = (1/2) m BD 6. mAMC = (1/2) m AC + (1/2) m BD 7. mAMC = mBMD = (1/2) (m AC + m BD)

O

A

BC

D

M12


Chord-Chord Angle ExampleIf O with chords AB, CD, AC and BD, calculate m AC and

m AD

O

A

BC

D

M70o

90o

60o


Chord-Chord Angle ExampleIf O with chords AB, CD, AC and BD, calculate m AC and

m AD

O

A

BC

D

M70o

90o

60o

80o

130o



Measure an angle formed by

A tangent and a chord

Two tangents

Two secants

A tangent and a secant

Two chords


Exercises


Exercise 1 O has a tangent ED and two parallel chords CD and AB. If the

inscribed angle DAB = 20o, Find CDE.

OBA

D

20o

E

C


Exercise 1 O has a tangent ED and two parallel chords CD and AB. If the

inscribed angle DAB = 20o, Find CDE.

OBA

D

20o

E

C

40o40o

100o

50o


Exercise 2 C has a tangent AB and a secant AE. If m BE = 120, m BD = ? m EF

= ? mA = ?

C

B

A

DE

F

120o


Exercise 2 C has a tangent AB and a secant AE. If m BE = 120, m BD = ? m EF

= ? mA = ?

C

B

A

30o

DE

F

60o

120o

60o

60o

60o

60o

30o


Exercise 3 O has two secants CA and CB. If AE = ED and mEAB = 65, find

ECB = ?

C

B

A

D

65o

O

E


Exercise 3 O has two secants CA and CB. If AE = ED and mEAB = 65, find

ECB = ?

C

B

A

D

65o

O

E

25o 25o

65o


Exercise 4 ABCDE is a regular pentagon inscribed in O and BG is a

tangent. Find ABG and AFE.C

B

A

D

O

E

FG


Exercise 4 ABCDE is a regular pentagon inscribed in O and BG is a

tangent. Find ABG and AFE.C

B

A

D

O

E

FG

72o

36o

72o72o

108o


Segment Measurement Theorems


Segment Measurement Theorems

Measure segments formed by

Two chords

A secant and a tangent

Two secants


Segments Formed by Two Chords


Chord-Chord Segment TheoremIf two chords intersect within a circle, the product of the

measures of the segments of one chord equals the product of the measures of the segments of the other chord

Given: AB and CD are chords of O, two chords intersect at E

Prove: AE · BE = CE · DEO

A

B

C

DE


Chord-Chord Segment TheoremIf two chords intersect within a circle, the product of the

measures of the segments of one chord equals the product of the measures of the segments of the other chord

Given: AB and CD are chords of O, two chords intersect at E

Prove: AE · BE = CE · DEO

A

B

C

DE

1

2 3

4


Chord-Chord Segment TheoremGiven: AB and CD are chords of O,

two chords intersect at E Prove: AE · BE = CE · DE1. Connect BC and AD

2. m1 = m2 (Congruent inscribed angles)

3. m3 = m4 (Congruent inscribed angles)

4. CBE ~ ADE (AA similarity)

5. AE/CE = DE/BE (Corresponding sides proportional)

6. AE · BE = CE · DE (Cross product)

O

A

B

C

DE

1

2 3

4


Chord-Chord Segment ExampleIf O with chords AB and CD, CD = 10, CM = 6, and AM = 8,

calculate AB = ?

O

A

BC

D

M

6

8

10


Chord-Chord Segment ExampleIf O with chords AB and CD, CD = 10, CM = 6, and AM = 8,

calculate AB = ?

AM · BM = CM · DM

8 · BM = 6 · (10 - 6)

BM = 24 / 8 = 3

AB = 3 + 8 = 11

O

A

BC

D

M

6

8

103

4


Segments Formed by A Secant and A

Tangent


Secant-Tangent Segment TheoremIf a tangent and a secant are drawn to a circle from the same

external point, then length of the tangent is the mean proportional between the lengths of the secant and its external segment

Given: A is an external point to O, AD is a secant and AC is a tangent of O,

Prove: AD · AB = AC2

A

C

B

O

D


Secant-Tangent Segment TheoremIf a tangent and a secant are drawn to a circle from the same

external point, then length of the tangent is the mean proportional between the lengths of the secant and its external segment

Given: A is an external point to O, AD is a secant and AC is a tangent of O,


A

C

B

O

D

2

1


Secant-Tangent Segment TheoremGiven: A is an external point to O,

AD is a secant and AC is a tangent of O,


1. Connect BC and CD

2. m1 = (1/2) m BC (Tangent-chord angles theorem)

3. m2 = (1/2) m BC (Inscribed angles theorem)

4. m1 = m2 (Substitution property)

5. mA = mA (Reflexive property)

6. CBA ~ DCA (AA similarity)

7. AB/AC = AC/AD (Corresponding sides proportional)

8. AD · AB = AC2 (Cross product)

A

C

B

O

D

2

1


Secant-Tangent Segment ExampleIf O with tangent AC and secant AD, OD = 5 and AB = 6,

calculate AC = ?

A

C

B O 56 D


Secant-Tangent Segment ExampleIf O with tangent AC and secant AD, OD = 5 and AB = 6,

calculate AC = ?

AB = 6

AD = 16

AC2 = AD · AB

AC2 = 16 · 6

AC = 4 √6

A

C

B O 56 D5

4 √6


Segments Formed by Two Secants


Secant-Secant Segment TheoremIf two secants are drawn to a circle from the same external

point then the product of the lengths of one secant and its external segment is equal to the product of the lengths of the other secant and its external segment

Given: A is an external point to O, AD and AE are secants

to OProve: AD · AB = AE · AC

A

CE

B

O

D


Secant-Secant Segment TheoremIf two secants are drawn to a circle from the same external

point then the product of the lengths of one secant and its external segment is equal to the product of the lengths of the other secant and its external segment

Given: A is an external point to O, AD and AE are secants

to OProve: AD · AB = AE · AC

A

CE

B

O

D

1

2


Secant-Secant Segment TheoremGiven: A is an external point to O,

AD and AE are secants to O

Prove: AD · AB = AE · AC

1. Connect BE and CD



4. m1 = m2 (Substitution property)

5. mA = mA (Reflexive property)

6. EBA ~ DCA (AA similarity)

7. AD/AE = AC/AB (Corresponding sides proportional)

8. AD · AB = AE · AC (Cross product)

A

CE

B

O

D

1

2


Secant-Secant Segment ExampleIf O with secants AC and AE, OC = DE = x, AD = 10 and AB =

8, calculate BC = ?

ACB O

10

8

DE

x

x


Secant-Secant Segment ExampleIf O with secants AC and AE, OC = DE = x, AD = 10 and AB =

8, calculate BC = ?

AC · AB = AE · AD

8 (2x + 8) = 10 (10 + x)

4 (2x + 8) = 5 (10 + x)

8x + 32 = 50 + 5x

3x = 18

X = 6

BC = 12

ACB O

10

8

DE

6

12


Exercises


Exercise 1 O has a tangent AF and two secants AC and AB. If AD = 3, CD = 9, and AE =

4, find AF = ? BE = ?

O

B

AD

E

C

3

4

9

F


Exercise 1 O has a tangent AF and two secants AC and AB. If AD = 3, CD = 9, and AE =

4, find AF = ? BE = ?

AF2 = AD · AC = 3 · (3 + 9)

AF2 = 36

AF = 6

AF2 = AB · AE

36 = 4 (BE + 4)

BE = 5

O

B

AD

6

E

C

3

4

9

F

5


Exercise 2

C has two chords AF and DE. If AP = 6 and PF = 2, EP = 3, and CM = 3, then CN = ?

C

A

DE

F

2

6

3M

N

3

P


Exercise 2

C has two chords AF and DE. If AP = 6 and PF = 2, EP = 3, and CM = 3, then CN = ?

AP · PF = DP · PE

6 · 2 = DP · 3

DP = 4

AC = 5

CN = (52 - 3.52)1/2

C

A

DE

F

2

6

34

M

N

3

P

5

5


Circles in a Coordinate Plane


Circles in a Coordinate PlaneA circle with center at the origin and a radius with a length of 5.

The points (5, 0), (0, 5), (-5, 0) and (0, -5) are points on the circle. What is the equation of the circle?

O

B (0, 5)

5

D (0, –5)

x

C (-5, 0)x

y

y P (x, y)


A (5, 0)y

x

Circles in a Coordinate PlaneA circle with center at the origin and a radius with a length of 5.


x2 + y2 = 52

or

x2 + y2 = 25 O

B (0, 5)

5

D (0, –5)

x

C (-5, 0)x

y

y P (x, y)


A (5, 0)y

x

Circles in a Coordinate PlaneA circle with center at the origin and a radius with a length of r.

The points (r, 0), (0, r), (-r, 0) and (0, -r) are points on the circle. What is the equation of the circle?

O

B (0, r)

r

D (0, –r)

x

C (-r, 0)x

y

y P (x, y)


A (r, 0)y

x

Circles in a Coordinate PlaneA circle with center at the origin and a radius with a length of r.

The points (r, 0), (0, r), (-r, 0) and (0, -r) are points on the circle. What is the equation of the circle?

x2 + y2 = r2

O

B (0, r)

r

D (0, –r)

x

C (-r, 0)x

y

y P (x, y)


A (r, 0)y

x

Circles in a Coordinate PlaneA circle with center at the (2, 4) and a radius with a length of 5.


(2, 4)

B (2, 9)

5

D (2, –1)

x

C (-3, 4)

y P (x, y)


A (7, 4)|y -4|

|x – 2|

y = 4

x = 2

Circles in a Coordinate PlaneA circle with center at the (2, 4) and a radius with a length of 5.


(x – 2)2 + (y – 4)2 = 52

or

(x – 2)2 + (y – 4)2 = 25 (2, 4)

B (2, 9)

5

D (2, –1)

x

C (-3, 4)

y = 4

x = 2

y P (x, y)


A (7, 4)|y -4|

|x – 2|

Circles in a Coordinate PlaneA circle with center at the (h, k) and a radius with a length of r.

The points (h+r, k), (h, k+r), (h-r, k) and (h, k-r) are points on the circle. What is the equation of the circle?

(h, k)

B (h, k+r)

r

D (h, k–r)

x

C (h-r, k)

y P (x, y)


A (h+r, k)|y -k|

|x–h|

y = k

x = h

Circles in a Coordinate PlaneA circle with center at the (h, k) and a radius with a length of r.

The points (h+r, k), (h, k+r), (h-r, k) and (h, k-r) are points on the circle. What is the equation of the circle?

(x – h)2 + (y – k)2 = r2

(h, k)

B (h, k+r)

r

D (h, k–r)

x

C (h-r, k)

y = k

x = h

y P (x, y)


A (h+r, k)|y -k|

|x–h|

Equation of a CircleCenter-radius equation of a circle with radius r and

center (h, k) is

(x – h)2 + (y – k)2 = r2

(h, k)

r

P (x, y)


Center of a CircleA circle has a diameter PQ with end-points at P (x1, y1) and

Q (x2, y2). What is the center C (h, k) of the circle?

C (h, k)r

P (x1, y1)


Q (x2, y2)

r

Center of a CircleA circle has a diameter PQ with end-points at P (x1, y1) and

Q (x2, y2). The center C (h, k) of the circle is the midpoint of the diameter

C (h, k) = ( , )

r

P (x1, y1)


Q (x2, y2)

r

x1 + x2 y1 + y2 2 2

C (h, k)

Center of a Circle ExampleA circle has a diameter PQ with end-points at P (5, 7) and Q

(-1, -1). Find the center of the circle, C (h, k)

r

P (5, 7)


Q (-1, -1)

r

C (h, k)

Center of a Circle ExampleA circle has a diameter PQ with end-points at P (5, 7) and Q

(-1, -1). Find the center of the circle, C (h, k)

C (h, k) = ( , )

= (2, 3)

r

P (5, 7)


Q (-1, -1)

r

5 + (-1) 7 + (-1) 2 2 C (h, k)

Radius of a CircleA circle has a diameter PQ with end-points at P (x1, y1) and

Q (x2, y2). What is the radius (r) of the circle?

C (h, k)r

P (x1, y1)


Q (x2, y2)

r

Radius of a CircleA circle has a diameter PQ with end-points at P (x1, y1) and

Q (x2, y2). The radius (r) of the circle is equal to ½ PQ

r = ½ PQ

= ½ √ (x2 – x1)2 + (y2 – y1)2

C (h, k)r

P (x1, y1)


Q (x2, y2)

r

Radius of a Circle ExampleA circle has a diameter PQ with end-points at P (5, 7) and Q

(-1, -1). What is the radius (r) of the circle?


r

P (5, 7)

Q (-1, -1)

r

C (h, k)

Radius of a Circle ExampleA circle has a diameter PQ with end-points at P (5, 7) and Q

(-1, -1). What is the radius (r) of the circle?

r = ½ PQ

= ½ √ (-1 – 5)2 + (-1 – 7)2

= ½ (10)

= 5


r

P (5, 7)

Q (-1, -1)

r

C (h, k)

Circles in a Coordinate Plane Exercise

(a) Write an equation of a circle with center at (3, -2) and radius of length 7

(b) What are the coordinates of the endpoints of the horizontal diameter?



(a) Write an equation of a circle with center at (3, -2) and radius of length 7 (x–3)2 + (y+2)2 = 49

(b) What are the coordinates of the endpoints of the horizontal diameter? (10, -2), (-4, -2)



A circle C has a diameter PQ with end-points at P (-2, 9) and Q (4, 1)

(a) What is the center (C) of the circle?

(b) What is the radius (r) of the circle?

(c) What is the equation of the circle?

(d) What are the coordinates of the endpoints of the horizontal diameter?

(e) What are the coordinates of the endpoints of the vertical diameter?

(f) What are the coordinates of two other points on the circle?



A circle C has a diameter PQ with end-points at P (-2, 9) and Q (4, 1)

(a) What is the center (C) of the circle? (1, 5)

(b) What is the radius (r) of the circle? 5

(c) What is the equation of the circle? (x–1)2 + (y–5)2 = 25

(d) What are the coordinates of the endpoints of the horizontal diameter? (-4, 5), (6, 5)

(e) What are the coordinates of the endpoints of the vertical diameter? (1, 10), (1, 0)

(f) What are the coordinates of two other points on the circle?

(4, 9), (-2, 1)



Based on the diagram,

(a) write an equation of the circle

(b) Find the area of the circle



Based on the diagram,

(a) write an equation of the circle

(b) Find the area of the circle

(a) (x+4)2 + (y+4)2 = 25

(b) 25π


Q & A


The End


eleanor roosevelt high school chin-sung lin. mr. chin-sung lin erhs math geometry

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