CirclesCirclesVocabulary

AndProperties

VocabularyAnd

Properties

CircleCircleA set of all points in a plane at a

given distance (radius) from a given point (center) in the plane.

A set of all points in a plane at a given distance (radius) from a given point (center) in the plane.

r

center

RadiusRadius

A segment from a point on the circle to the center of the circle.

A segment from a point on the circle to the center of the circle.

r

Congruent CirclesCongruent CirclesTwo circles whose radii have the same

measure.Two circles whose radii have the same

measure.

r =3 cm r =3 cm

Concentric CirclesConcentric Circles

Two or more circles that share the same center.Two or more circles that share the same center.

.

ChordChord

A segment whose endpoints lie on the circle.Segments AB & CD are chords of G

A segment whose endpoints lie on the circle.Segments AB & CD are chords of G

AB

DC

G

DiameterDiameter

A chord passing through the center of a circle.Segment IJ is a diameter of G

A chord passing through the center of a circle.Segment IJ is a diameter of G

I

J

G

SecantSecant

A line that passes through two points of the circle.

A line that contains a chord.

A line that passes through two points of the circle.

A line that contains a chord.

TangentTangent

A line in the plane of the circle that intersects the circle in exactly one point.

A line in the plane of the circle that intersects the circle in exactly one point.

●

●

The point of contact is called the Point of Tangency

The point of contact is called the Point of Tangency

SemicircleSemicircle

A semicircle is an arc of a circle whose endpoints are the endpoints of the diameter.

A semicircle is an arc of a circle whose endpoints are the endpoints of the diameter.

is a semicircle

C

BA

●

ACB

Three letters are required to name a

semicircle: the endpoints and one

point it passes through.

Minor ArcMinor Arc

An arc of a circle that is smaller than a semicircle. An arc of a circle that is smaller than a semicircle.

P

C

B

●

PC or CB are minor arcs

Two letters are required to name a minor arc:

the endpoints.

Major ArcMajor Arc

An arc of a circle that is larger than a semicircle. An arc of a circle that is larger than a semicircle.

C

BA

●

ABC or CAB are major arcs

Inscribed AngleInscribed Angle

An angle whose vertex lies on a circle and whose sides contain chords of a circle.

An angle whose vertex lies on a circle and whose sides contain chords of a circle.

B

A

C

D

<ABC & <BCD are inscribed angles

Central AngleCentral Angle

An angle whose vertex is the center of the circle and sides are radii of the circle.

An angle whose vertex is the center of the circle and sides are radii of the circle.

A

KB

<AKB is a central angle

Properties of CirclesProperties of Circles

The measure of a central angle is two times the measure of the inscribed angle

that intercepts the same arc.

The measure of a central angle is two times the measure of the inscribed angle

that intercepts the same arc.

P

AB

Cm<APB = 2 times m<ACB

½ m<APB = m<ACB

x

2x

ExampleExampleIf the m<C is 55, then the m<O is 110.

Both angle C and angle O intercept the same arc, AB.

If the m<C is 55, then the m<O is 110. Both angle C and angle O intercept the same arc, AB.

O

AB

C

55°

110°

Angles inscribed in the same arc are congruent.Angles inscribed in the same arc are congruent.

A

Q

B

P

m<QAP = m<PBQBoth angles intercept QP

The m<AQB =m<APB both intercept arc AB.

Every angle inscribed in a semicircle is an right angle.

Every angle inscribed in a semicircle is an right angle.

ExampleExample

Each of the three angles inscribed in the semicircle is a right angle.

Each of the three angles inscribed in the semicircle is a right angle.

A

B

C D

E Angle B, C, and D are all 90 degree angles.

Property #4Property #4

The opposite angles of a quadrilateral inscribed in a circle are supplementary.

The opposite angles of a quadrilateral inscribed in a circle are supplementary.

ExampleExample

The measure of angle D + angle B=180The measure of angle C+angle A=180The measure of angle D + angle B=180The measure of angle C+angle A=180

A

B

C

D

110

70

115

65

Property #5Property #5

Parallel lines intercept congruent arcs on a circle.

Parallel lines intercept congruent arcs on a circle.

ExampleExample

A

B

Arc AB is congruent to Arc CDArc AB is congruent to Arc CD

C

D

FormulasFormulas

What are the two formulas for finding circumference?

C=C=

What are the two formulas for finding circumference?

C=C=

AnswerAnswer

C=2 pi r

C=d pi

C=2 pi r

C=d pi

Area of a circleArea of a circle

A=?A=?

AnswerAnswer

A=radius square times piA=radius square times pi

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