Circles

Chapter 12

Parts of a Circle

A

•

•

••

M

•

•

B

C

D

E

Chord

diameter

radius

Circle: The set of all points in a plane that are a given distance from a given point in that plane.

Center: The middle of the circle – a circle is named by its center, the

symbol of a circle looks like - ּסM.

Radius: a segment that has one endpoint at the center and the other endpoint on the circle. The radius is ½ the length of the diameter. ALL RADII ARE CONGRUENT.

More Vocabulary

A

•

•

••

M

•

•

B

C

D

E

Chord

diameter

radius

Chord: a segment that has its endpoints on the circle.

Diameter: a chord that passes through the center of the circle. The diameter is 2 times the radius.

Circumference: the distance around the circle. To find the circumference use: rC 2

Arcs: the space on the circle between the two points on the circle.

Tangent Lines

AB

•

•

•

T

P

•

A

B

Tangent line: A line that intersects the circle at exactly one point. is a tangent line to ּסT.

Point of Tangency: the point where the circle and the tangent line intersect.

Theorem 12-1: If a line is tangent to a circle, then the line is perpendicular to a radius drawn to the point of tangency.

TPAB

Tangent Lines Continued

•

•X

•

•

P

Q

R

Theorem 12-3: Two segments tangent to a circle from the same point outside of the circle are congruent.

QRPQ

Central Angles

DGDF

DGDF

Central Angles: angles whose vertex is the center of the circle.

Theorem 12-4: Within a circle or congruent circles:

(1)Congruent central angles have congruent chords.

(2)Congruent chords have congruent arcs.

(3)Congruent arcs have congruent central angles.

•

•

•

D

E

F

•G

37º37º

DEFGED

Chords

CDAB

Theorem 12-5:Within a circle or congruent circles

(1) Chords equidistant from the center are congruent.

(2) Congruent chords are equidistant from the center.

•P

A • • •

••

•

E B

C

F

D

PFEP

More About Chords

VSUV RSUR

Theorem 12-6:In a circle, a diameter that is perpendicular to a chord bisects the chord and its arc.

Theorem 12-7:In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord.

Theorem 12-8:In a circle, the perpendicular bisector of a chord contains the center of the circle.

•T

•

•

•

V

U

R

S

Q • •

USQR

Inscribed Angles

Inscribed Angle: An angle whose vertex is on the circle, and the sides are chords of the circle.

Intercepted Arc: an arc of a circle having endpoints on the sides of an inscribed angle. AB

•D

•

••

A

B

CC

mACBm2

1

Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Corollaries:1.Two inscribed angles that intercept the same arc are congruent.2.An inscribed angle in a semicircle is a right angle.3.The opposite angles of a quadrilateral inscribed in a circle are supplementary.

●

●

●

A

B

C

90° 45°

1

38° 0381

●

●

● ●2

0902m

1

23 4

018031 mm

An angle formed by a tangent line and a chord.

The measure of angle formed by a tangent line and a chord is half the measure of the intercepted arc.

mBDCCm2

1

●

●

●

●

C

D

B

65°

130°

Secant Line

A secant line is a line that intersects a circle at two points.

●

●

●Theorem 12-11:The measure of an angle formed by two lines that(1)Intersect inside a circle is half the sum of the measure of the intercepted arcs.

(2)Intersect outside the circle is half the difference of the measures of the intercepted arcs.

1x° y°

yxm 2

11

yxm 2

11

●

1

x°

y°

Segment Length Theorems

1. a ● b = c ● d

2. (w + x)w = (y + z)y

3. (y + z)y = t2

a

b

c

d●

w

x

yz

●

t

yz

●

Equation of a Circle

An equation of a circle with the center (h, k) and radius r is (x – h)2 + (y – k)2 = r2.

Example: Center (5, 3) radius 4.(x – 5)2 + (y – 3)2 = 16