Post on 19-Jan-2016
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