A radius drawn to a tangent at the point of tangency is

perpendicular to the tangent.

l

C

T

Line l is tangent to Circle

C at point T.

CT l at T

Two tangent segments drawn to a circle from the same

exterior point are congruent.A

C

B

O

CA and CB are

tangent segments

.

CA CB

1 2

3

4

5

O

A

B

CENTRALANGLE

--VERTEX IS AT THE CENTER--SIDES ARE

RADII--ITS MEASURE IS

EQUAL TO THE MEASURE OF ITS

INTERCEPTED ARC

1

m AOB mAB

O

A

B

1

The arc of an

angle is the

portion of the circle

in the INTERIOR

of the angle.

O

A

BC

INSCRIBEDANGLE

--VERTEX IS ON THE CIRCLE--SIDES ARE CHORDS

--ITS MEASURE IS EQUAL TO HALF THE

MEASURE OF ITS INTERCEPTED ARC

12

m ACB mAB

ANGLE FORMED BY

CHORD(SECANT) AND TANGENT

WITH VERTEX ON CIRCLE

--ITS MEASURE IS EQUAL TO HALF

THE MEASURE OF ITS INTERCEPTED

ARC

C 3O

A

B12

m CAB mAB

ANGLE FORMED BY TWO CHORDS

(SECANTS) WHOSE VERTEX IS IN THE INTERIOR OF THE

CIRCLE, BUT NOT AT THE CENTER

--ITS MEASURE IS EQUAL TO HALF THE SUM OF THE MEASURES OF ITS INTERCEPTED

ARCS (ITS ARC AND THE ARC OF ITS VERTICAL ANGLE)

C

A

B

D

E

1( )2

m CAB mCB DE

AN ANGLE WHOSE VERTEX

IS IN THE EXTERIOR

OF A CIRCLE MAY BE FORMED

BY:

A TANGENT AND A

SECANT

TWO SECANTS

ORTWO

TANGENTS

B

ACD

BA

C

D

A

B

C

D

E

IN EACH CASE, THE MEASURE OF

THE ANGLE IS ONE-HALF THE DIFFERENCE OF

THE MEASURES OF THE INTERCEPTED

ARCS.

B

ACD

A

B

C

D

BA

C

DE

1( )2

m A mBCD mBD

1( )2

m A mCD mBE

1( )2

m A mBC mBD

Two points on a circle will always

determine:

A CHORD

TWO RADII

A MINOR ARC

A MAJOR ARC

OR TWO SEMICIRCLES

CONGRUENT CENTRAL ANGLES WILL HAVE

CONGRUENT CHORDS AND ARCS (AND VICE-

VERSA) IF AND ONLY IF THEY ARE IN THE SAME

OR IN CONGRUENT CIRCLES!!

IN THE SAME CIRCLE OR IN CONGRUENT

CIRCLES, CONGRUENT CHORDS ARE EQUIDISTANT

FROM THE CENTER.

IN THE SAME CIRCLE OR IN CONGRUENT CIRCLES, IF TWO

CHORDS ARE EQUIDISTANT FROM THE CENTER, THEN

THEY ARE CONGRUENT.

IN THE SAME CIRCLE OR IN CONGRUENT CIRCLES, IF TWO

CHORDS ARE UNEQUAL, THEN THE LONGER CHORD IS

CLOSER TO THE CENTER.

IN THE SAME CIRCLE OR IN CONGRUENT

CIRCLES, IF THE DISTANCES FROM THE

CENTER OF TWO CHORDS ARE UNEQUAL,

THEN THE LONGER CHORD IS CLOSER TO

THE CENTER.

O

“Anything” from the center of a circle

(segment,

radius,

diameter)

A B

O

perpendicular to a chord

A B

O

A B

bisects “everything” it touches:

O

A B

the chord

C

O

A B

the central angle

C

O

A B

the minor arc

C

O

A B

the major arc

C

Describe how the center,O, can be located.

.AB and CD are chords of O

BA

C

D

Construct the perpendicular bisectors of

the chords. They will intersect at the center!B

AC

D

O

A

B

12

IF TWO INSCRIBED

ANGLES INTERCEPT THE SAME ARC, THEN THEY ARE

CONGRUENT.

B

OA C

D

IF AN ANGLE IS

INSCRIBED IN A

SEMICIRCLE,THEN IT IS

A RIGHT ANGLE.

A

BC

D

O

IF A QUADRILATERAL

IS INSCRIBED IN A CIRCLE, THEN ANY PAIR OF OPPOSITE

ANGLES ARE SUPPLEMENTARY.

abc

da

bc

b

a

c

d

ab cd

2 a bc

ab dc

IF TWO CHORDS OF A CIRCLE INTERSECT, THEN THE PRODUCT OF THE LENGTHS OF THE SEGMENTS ON

ONE CHORD EQUALS THE PRODUCT OF THE LENGTHS OF THE SEGMENTS ON

THE OTHER CHORD. a

bc

dab = cd

IF A TANGENT SEGMENT AND A SECANT SEGMENT INTERSECT IN THE EXTERIOR OF A CIRCLE, THEN THE SQUARE OF THE LENGTH

OF THE TANGENT SEGMENT IS EQUAL TO THE PRODUCT OF THE LENGTHS OF THE

SECANT SEGMENT AND ITS EXTERNAL PART.

a

bc

a2 = bc

IF TWO SECANT SEGMENTS INTERSECT IN THE EXTERIOR OF A CIRCLE, THEN THE

PRODUCT OF THE LENGTHS OF ONE SECANT SEGMENT AND ITS EXTERNAL

PART IS EQUAL TO THE PRODUCT OF THE LENGTHS OF THE OTHER SECANT

SEGMENT AND ITS EXTERNAL PART.

a

b

dc

ab = cd