Geometry – Arcs, Central Angles, and Chords

An arc is part of a circle. There are three types you need to understand:

P

Semicircle – exactly half of a circle

180°

X

BA

Geometry – Arcs, Central Angles, and Chords

An arc is part of a circle. There are three types you need to understand:

P

P

D

C

Minor arc – less than a semicircle ( < 180° )

Semicircle – exactly half of a circle

180°

X

BA

Geometry – Arcs, Central Angles, and Chords

An arc is part of a circle. There are three types you need to understand:

P

P

D

C

Minor arc – less than a semicircle ( < 180° )

Semicircle – exactly half of a circle

180°

P

Major arc – bigger than a semicircle

( > 180° )

X

BA

A

B

E

Geometry – Arcs, Central Angles, and Chords

An arc is part of a circle. There are three types you need to understand:

P

P

D

C

Minor arc – less than a semicircle ( < 180° )

Semicircle – exactly half of a circle

180°

P

Major arc – bigger than a semicircle

( > 180° )

X

BA

A

B

E

The symbol for an arc ( ) is placed above the letters naming the arc

Geometry – Arcs, Central Angles, and Chords

An arc is part of a circle. There are three types you need to understand:

P

P

D

C

Minor arc – less than a semicircle ( < 180° )

Semicircle – exactly half of a circle

180°

P

Major arc – bigger than a semicircle

( > 180° )

X

BA

A

B

E

The symbol for an arc ( ) is placed above the letters naming the arc

AXBYou need 3 letters to name a semicircle

Geometry – Arcs, Central Angles, and Chords

An arc is part of a circle. There are three types you need to understand:

P

P

D

C

Minor arc – less than a semicircle ( < 180° )

- Use the ray endpoints to name a minor arc

Semicircle – exactly half of a circle

180°

P

Major arc – bigger than a semicircle

( > 180° )

X

BA

A

B

E

The symbol for an arc ( ) is placed above the letters naming the arc

AXBYou need 3 letters to name a semicircle

CD

Geometry – Arcs, Central Angles, and Chords

An arc is part of a circle. There are three types you need to understand:

P

P

D

C

Minor arc – less than a semicircle ( < 180° )

- Use the ray endpoints to name a minor arc

Semicircle – exactly half of a circle

180°

P

Major arc – bigger than a semicircle

( > 180° )

- Use the ray endpoints and a point in between to name a major arc

X

BA

A

B

E

The symbol for an arc ( ) is placed above the letters naming the arc

AXBYou need 3 letters to name a semicircle

CD

BEA

Geometry – Arcs, Central Angles, and Chords

A central angle is an angle whose vertex is at the center of a circle:

P

D

CDPC

Geometry – Arcs, Central Angles, and Chords

A central angle is an angle whose vertex is at the center of a circle:

P

D

CDPC

- This central angle creates an arc that is equal to the measure of the central angle.

CD

40 CD arc ,40 If DPCm

Geometry – Arcs, Central Angles, and Chords

A central angle is an angle whose vertex is at the center of a circle:

P

D

CDPC

-This central angle creates an arc that is equal to the measure of the central angle

CD

40 CD arc ,40 If DPCm

The reverse is also true, if arc CD = 50°, central angle DPC = 50°

Geometry – Arcs, Central Angles, and Chords

P

D

C

Chord DC separates circle P into two arcs, minor arc DC, and major arc DYC.

Y

Geometry – Arcs, Central Angles, and Chords

P

D

C

Theorem : if two chords of a circle have the same length, their intercepted arcs have the same measure.

A

B

Geometry – Arcs, Central Angles, and Chords

P

D

C

Theorem : if two chords of a circle have the same length, their intercepted arcs have the same measure.

A

B

ABDC then , If mmABDC

Geometry – Arcs, Central Angles, and Chords

P

D

C

Theorem : if two chords of a circle have the same length, their intercepted arcs have the same measure.

A

B

ABDC then , If mmABDC - The reverse is then also true, if intercepted arcs have the same measure, their chord have the same length.

Geometry – Arcs, Central Angles, and Chords

P

D

C

Theorem : if two chords of a circle have the same length, their intercepted arcs have the same measure.

A

B

ABDC then , If mmABDC - The reverse is then also true, if intercepted arcs have the same measure, their chord have the same length.

EXAMPLE : CD = AB and the measure of arc AB = 86°. What is the measure of arc CD ?

Geometry – Arcs, Central Angles, and Chords

P

D

C

Theorem : if two chords of a circle have the same length, their intercepted arcs have the same measure.

A

B

ABDC then , If mmABDC - The reverse is then also true, if intercepted arcs have the same measure, their chord have the same length.

EXAMPLE : CD = AB and the measure of arc AB = 86°. What is the measure of arc CD ?

86CD

Geometry – Arcs, Central Angles, and Chords

P

DC

Theorem : chords that are equidistant from the center have equal measure

A

B

X

Y

ABCD then ,YPXP If

Geometry – Arcs, Central Angles, and Chords

P

DC

Theorem : chords that are equidistant from the center have equal measure

A

B

X

Y

ABCD then ,YPXP If

EXAMPLE : XP = YP and the measure of AB = 30. What is the measure of CD ?

Geometry – Arcs, Central Angles, and Chords

P

DC

Theorem : chords that are equidistant from the center have equal measure

A

B

X

Y

ABCD then ,YPXP If

EXAMPLE : XP = YP and the measure of AB = 30. What is the measure of CD ?

30 CD

Geometry – Arcs, Central Angles, and Chords

P

Theorem : If a diameter or radius is perpendicular to a chord, it bisects that chord and its arc.

AB

Y

X

BY AY and XBAX then AB, PY If

Geometry – Arcs, Central Angles, and Chords

P

Theorem : If a diameter or radius is perpendicular to a chord, it bisects that chord and its arc.

AB

Y

X

BY AY and XBAX then AB, PY If

EXAMPLE : PY is perpendicular to and bisects AB, arc AB = 100°.

What is the measure of arc YB ?

Geometry – Arcs, Central Angles, and Chords

P

Theorem : If a diameter or radius is perpendicular to a chord, it bisects that chord and its arc.

AB

Y

X

BY AY and XBAX then AB, PY If

EXAMPLE : PY is perpendicular to and bisects AB, arc AB = 100°.

What is the measure of arc YB ?

50YB