10.2 Arcs and chords10.2 Arcs and chords

Pg 603

Central angleCentral angle

• Central angle- angle whose vertex is the center of a circle

A

B

C ACB is ACB is a central a central angleangle

ArcsArcs

• Arc- a piece of a circle.

Named with 2 or 3 letters

Measured in degrees

• Minor arc- part of a circle that measures less than 180o (named by 2 letters).

A

B

B

P

BP

(

More arcsMore arcs

• Major arc- part of a circle that measures between 180o and 360o.

(needs three letters to name)

• Semicircle- an arc whose endpts are the endpts of a diameter of the circle

(OR ½ of a circle)

A

B

C

C

S

ABC or CBA

( (

Arc measuresArc measures

• Measure of a minor arc- measure of its central

• Measure of a major arc- 360o minus measure of minor arc

Ex: find the arc measuresEx: find the arc measures

A

B

C

D

E

50o

m AB=

m BC=

m AEC=

m BCA=

50o

130o

180o

180o+130o = 310o130o

180o(

((

(

OR 360o- 50o = 310o

Post. 26Post. 26arc addition postulatearc addition postulate

• The measure of an arc formed by two adjacent arcs is the sum of the measures of those arcs.

AB

Cm ABC = m AB+ m BC

( ( (

Congruency among arcsCongruency among arcs

• Congruent arcs- 2 arcs with the same measure

• MUST be from the same circle OR circles!!!

ExampleExample

30o

30o

A

B

C

DE

m AB=30o

m DC=30o

AB DC

((

( (

Ex: continuedEx: continued

A

B

C

DE

45o

m BD= 45o

m AE= 45o

BD AE

((

( (

The arcs are the same measure; so, why aren’t they ?

The 2 circles are NOT !

Thm 10.4Thm 10.4• In the same circle (or in circles), 2 minor

arcs are iff their corresponding chords are . A

B

C

AB BC iff AB BC

( (

Thm 10.5Thm 10.5• If a diameter of a circle is to a chord,

then the diameter bisects the chord and its arc.

E

DG

CF

If EG is to DF, then DC CF and DG GF

( (

If JK is a bisector of ML, then JK is a diameter.

Thm 10.6Thm 10.6• If one chord is a bisector of another

chord then the 1st chord is a diameter.

JK

L

M

Ex: find m BCEx: find m BC

A

B

CD

3x+11 2x+47

By thm 10.4 BD BC.

3x+11=2x+47x=36

2(36)+4772+47119o

(( (

Thm10.7Thm10.7• In the same circle (or in circles), 2 chords

are iff they are =dist from the center.

D C

A

G F

E B

DE CB iff AG AF

Ex: find CG.Ex: find CG.

A

B

C

E

F

D

6

6

6

6

7

G72=CF2+62

49=CF2+3613=CF2

CF CG

CF = ð13

CG = ð13

Assignment