CIRCLES Kelompok 6 Asti Pujiningtyas 4101414009 Eva Wulansari 4101414023 Mifta Zuliyanti4101414016 Zuliyana Dewi A. 4101414001.

CIRCLESKelompok 6

• Asti Pujiningtyas 4101414009• Eva Wulansari 4101414023

• Mifta Zuliyanti4101414016• Zuliyana Dewi A. 4101414001

10-1 Basic Definitions

Definition 10-1A radius of a circle is a segment whose endpoints are the center and a point on the circle.

A

B

Radius

A

DC Definiton 10-2A chord of a circle is a segment with endpoints on the circle.

chord

A

Definition 10-3A diameter of a circle is a chord that contain the center of the circle.

Definition 10-4A tangent to a circle is a line that intersect the circle in exactly one point.

Definition 10-5A secant of a circle is a line that intersect the circle in exactly two points.

G H

lA

B

Diameter

A

D

E

m

Definition 10-6An inscribed angle is an angle with vertex on a circle and with sides that contain chords of the circle.

Definition 10-7A central angle is an angle with vertex at the center of a circle.

A

H I

G

A

K

J

10-2 The Degree Measure of Arcs

Definition 10-8A minor arc is an arc that lies in the interior of a central angle. Otherwise ut is called a major arc

Definition 10-9The measure of a minor arc is the measure of its associated central angle. The measure of a major arc is 360 minus the measure of its associated minor arc.

O

A

B

Minor arc

Major arc

A A

B

70

Arc Addition PostulateIf C is on AB , then mAC + mCB = mAB

Definition 10-10If two arcs of a circle have the same measure, they are called congruent. If AB and CD are congruent, we write AB CD .

Definition 10-11Two circles are congruent if they have radii of equal lenght.A

B

DC

50°

50°

D

C

B

A

These two figures should focus your attention on the relationship between congruent chords and their arcs

Given congruent chords CDAB

Given congruent AB CD

A

B

C

D

A

B

C

D

A

B

C

D

Statement Reason

1. Given

2. OA=OB=OC=OD Definition of Circle

3. Definition of congruent segment

4. ∆OAB ∆OCD SSS Postulate

5. CPCTC

CDAB

ODOCOBOA

Theorem 10-1In a circle or in congruent circles congruent chords have congruent minor arcs.

CODAOB

O

Theorem 10-2In a circle or in congruent circles congruent minor arcs have congruent chords.

A

B

C

D

Statement Reason

1. AB CD Given

2. OA=OB=OC=OD Definitoin of Circle

3. Definition of congruent segment

4. SAS Postulate

5. CPCTC

ODOCOBOA

OCDOAB

O

CDAB

In each figure a pair of congruent chords is given.

In each case does XL = XM?These examples suggest the following theorem.

10-3. Chords and Distances from the Center

Theorem 10-3. In a circle or in congruent circles congruent chords are equidistant from the

center.

PROOFGiven : circle O, , , Prove : OM = OL

CDAB ABOM CDOL

Statements Reasons

1. 1. Given

2. OA = OB = OC = OD 2. Definition of circle

3. 3. Definition of congruent segments

4. 4. SSS congruence

5. 5. CPCTC6. and 6. Given

7. , and are right

angles.

7. Perpendicular lines from congruent right angles

8. and are right triangles

8. Definition of right triangles

9. 9. HA Congruence

10. 10. CPCTC

11. OM = OL 11. Definition of congruent segments

CODAOB 21

OLDOMB OMB

OLD

OMB OLD

OLOM

OLDOMB

CDAB

ODOCOBOA

ABOM CDOL

Theorem 10-4. In a circle or in congruent circles chords equidistant from the center are

congruent

PROOFGiven : ʘO, OM = OL, and Prove : CDAB

ABOM CDOL

Statements Reasons

1. OM = OL 1. Given

2. 2. Definition of congruence segment

3. OA=OB=OC=OD 3. Definition of circle

4. 4. Definitions of congruence segment

5. and 5. Given

6. 6. HL theorem

7. 7. CPCTC

8. CL=LD=AM=MB 8. Definition of congruence segment

9. AB = CD 9. Definition of congruence segment

10. 10. Definition of congruence segment

OLOM

MBAMLDCL

CDAB

MOBAOM

OCLODL

ODOCOBOA

ABOM CDOL

Perpendicular to Chords

Theorem 10.5 The perpendicular bisector of a chord contain the center of the circle

PROOF:Given: is a chord of circle O, and l is the perpendicular bisector of Prove: O is a point of l

AB

AB

O

B

A

l

Statement Reason

1. l is the perpendicular bisector of

2. OA = OB3. O lies on l

1. Given

2. Definition of circle3. A point equidistant

from point A and B belongs to the perpendicular bisector of (Theorem 6-10)

AB

AB

O

B

A

l

APPLICATIONFind the center of around table.Step 1 Select any two chords, and Step 2 Draw the perpendicular bisector p of , and perpendicular bisector q of .Conclusion:

By the Theorem 10-5 the center lies on both lines p and q. Consequently, the center of the table must be the intersection of these lines.

AB

CD

AB

CD

O

A

B

p

D

C

q

Theorem 10.6If a line through the center of a circle is

perpendicular to a chord that is not diameter, then it bisects the chord and its minor arc.

O

A

B

C

Statement Reasons

1. 2. OB = OA 3.

4. 5. ∆OCB ∆OCA6. 7. BC = CA

8. 9. AC BC

1. Given2. Definition of Circle3. Definition of congruent

segment4. Reflective property5. HL Theorem6. CPCTC7. Definition of congruent

segments8. CPCTC9. Definition 10-10

OAOB

OCOC

CABC

OCAOCB

O

A

B

C

AOCBOC

Theorem 10.7If a line through the center of a circle bisects a

chord that is not a diameter, then its perpendicular to the chord

O

A

B

C

Statement Reasons

1. 2. 3. 4. ∆OCB ∆OCA5. 6.

1. Given2. Definition of Circle3. Reflective property4. SSS Postulate5. CPCTC6. Perpendicular lines from

congruent right angles

OAOB OCOC

BCAC

O

A

B

C

OCBOCA

90 OCAmOCAm

10-5 Tangents to Circles• A line is tangent to a circle if it intersects

the circle in exactly one point.

A

ℓ

.O

Theorem 10 – 8 If a line is perpendicular to a radius at a point on the circle, then the line is tangent to the circle.

PROOFGiven : ℓProve : ℓ is tangent to the circle.Plan : use an indirect proof. Assume ℓ is not

tangent to the circle. This means ℓ does not intersect the circle or ℓ intersects the circle in two places

OA

A ℓ

.O

ℓA

.O

B

Statements Reasons

1. ℓ intersects the circle at a second point B.

1. Indirect proof assumption

2. ℓ 2. Given

3. is a hypotenuse of a right triangle. 3. Definition of hypotenuse.

4. OB > OA 4. Length of the hypotenuse is greater than the length of either side.

5. OB = OA 5. Definition of circle.

OA

OB

A ℓ

.O

Statements 4 and 5 are contradictory. Hence the assumption is false and the line ℓ is tangent to the circle.

ℓA

.O

B ℓA

.O

B

Theorem 10 – 9If a line is tangent to a circle, then the radius drawn to the point of contact is perpendicular to the tangent.

PROOFGiven : Circle O with radius and tangent line .Prove : Plan : use an indirect proof. Assume is not .

OC

AB

ABOC ABOC

.O

C..BA A

.O

DE B..

C

Statements Reasons

1. is not 1. Indirect proof assumption

2. is tangent to the circle. 2. Given

3. Draw a point D on such that 3. Construction

4. Draw a point E on such that CD = DE and E is on different side of D.

4. Construction

5. , and

are right angles .

5. Perpendicular lines from congruent right angles

6. OD = OD 6. A segment is congruent to itself (reflexive property)

7. 7. SAS Postulate

8. OC = OE 8. CPCTC

ABOD

intersect the circle at two different points, so is not tangential to the circle. Hence the assumption is false and the radius is perpendicular to tangent .

AB

ABAB

OC

AB

ODEODC ODCODE

ODEODC

AB ABOC

AB

Theorem 10 – 10If a line is perpendicular to a tangent at a point on the circle, then the line contains the center of the circle.

PROOFGiven : is a tangent of circle O and ℓ is the

perpendicular of .Prove : O is a point of ℓ.

ABAB

.

C

O

.BA

.

ℓ

Statement Reason

1. ℓ Given

2. ℓ does not contain point O Indirect proof assumption

3. Draw radius from O to point C

Construction

4. Theorem 10-9

AB

ABOC There is exactly one line through C that perpendicular to , so line ℓ contain the center of the circle

AB