Circles

Chapter 9

Tangent Lines (9-1)

• A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point.

• The point where a circle and a tangent intersect is the point of tangency.

P

B

A

Tangent Lines (9-2)

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

P

B

A

Tangent Lines (9-2)

• Converse: If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

84

7

A

BD

P

B

A

Tangent Lines (9-2)

• Corollary: The two segments tangent to a circle from a point outside the circle are congruent.

• AB = BC

B

Q

A

C

Tangent Lines (9-1)

• “Inscribed in the circle ”

• “Circumscribed about the circle”

R

S T

U

V

A

B

C

Tangent Lines (9-1)

• Circle G is inscribed in quadrilateral CDEF. Find the perimeter of CDEF.

6 ft

11 ft

8 ft

7 ft

G

C

F

E

D

Arcs and Central Angles 9-3

• Central Angle (of a circle)- angle with its vertex at the center of the circle

• Arc- unbroken part of a circle• Minor Arc (less than 180 degrees)

• Name them using the endpoints

• Major Arc (more than 180 degrees)• Name them using three points

• Semicircles- two arcs formed by the endpoints of a diameter

Arcs and Central Angles 9-3

• Measure of a minor arc= measure of its central angle

• Measure of a major arc= 360 degrees – measure of its minor arc

• Adjacent arcs- arcs with exactly one point in common (crust of adjacent pizza slices)

Arc Addition Postulate

• The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs

• Similar to the Angle Addition Postulate

Congruent Arcs

• Arcs in the same circle or congruent circles

• Have equal measures• Arcs in two circles of different sizes

cannot be congruent, even if they have the same measure (to be congruent, they must be the same shape and size)

Theorem 9-3

• In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent

• STOP

Chords and Arcs (9-4)

• A chord is a segment whose endpoints are on a circle.

• Each chord cuts off a minor arc and a major arc

A

B

C

Chords and Arcs (9-4)

• Theorem: Within a circle or congruent circles

1.Congruent arcs have congruent chords.

2.Congruent chords have congruent arcs.

Chords and Arcs (9-4)

• Within a circle or in congruent circles…

A

D

C

B

Theorem 9-5• A diameter that is perpendicular to a

chord bisects the chord and its arc.Converse…• In a circle, a diameter that bisects a

chord (that is not the diameter) is perpendicular to the chord.

• Example

86 degrees

Chords and Arcs (9-3)

• Theorem: Within a circle or congruent circles

1.Chords equidistant from the center are congruent.

2.Congruent chords are equidistant from the center.

Chords and Arcs (9-4)

• Find x.

Chords and Arcs (9-4)

• Find HL and QJ.

• HL= 22, QJ = 4 √3

1126

JQ

H

L

Chords and Arcs (9-4)

• In a circle, the perpendicular bisector of a chord contains the center of the circle.

• STOP

Inscribed Angles (9-5)

• Inscribed angle – vertex on the circle, sides of angle are chords of circle

• Intercepted arc – arc formed when the sides of the inscribed angle cross the circle

A

B

C

Inscribed Angles (9-5)

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

A

B

C

Inscribed Angles (9-5)

• Find x and y.

• x= ½ *(80+70)• x= 75°• m arc BC= 360- (80+70+90)

= 120°

• y= ½ * (70+120)= 95°

7080

90

y

x

A

B

C

D

Inscribed Angles (9-5)

• Corollary- Two inscribed angles that intercept the same arc are congruent.

Inscribed Angles (9-5)

Corollary- An angle inscribed in a semicircle is a right angle.

• GeoGebra example

Inscribed Angles (9-5)

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

Inscribed Angles (9-5)

• Find the value of a and b.

• a= 90°

• 2 *32° = 64°

• b= 180- 64= 116°

a

b

32

E

• 9-5 handout• Problems 1-9 all

Inscribed Angles (9-5)

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

F

G

H

I

• 9-5 handout• Problems 10-21 all

Angle Measure and Segment Lengths (9-

5)• A secant is a line that intersects a

circle at two points.

B

A

Angle Measure and Segment Lengths (9-

6)• The measure of an angle formed

by two lines that intersect1.inside a circle is half the sum of

the measures of the intercepted arcs.

2.outside a circle is half the difference of the measure of the intercepted arcs.

The measure of an angle formed by two lines that intersect

inside a circle is half the sum of the measure of the intercepted arcs

• Find the measure of <1• m<1= ½ (45 + 75)• = 60

The measure of an angle formed by two lines that

intersectoutside a circle is half the

difference of the measure of the intercepted arcs.

• m <B = ½ (m AFD - m AC)• 65 = ½ (m AFD – 70)• 200 = m AFD

Angle Measure and Segment Lengths (9-

6)• Find the value of x.• x = ½ (268 – 92)• x = 88

268

92x

Angle Measure and Segment Lengths (9-

6)• Find the value of x.• 94 = ½ (x + 122)• 188 = x + 122• x = 66

94

112

x

Angle Measure and Segment Lengths (9-

6)Where the

angle vertex isAngle measure

Center of circle m(arc)

On circle ½ m(arc)

Inside circle ½ sum of m(arcs)

Outside circle ½ difference of m(arcs)

Angle Measure and Segment Lengths (9-

7)

(y + z)y = t2

t

y

z

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

w

z

y

x

a b = c dc

d

b

a

Angle Measure and Segment Lengths (9-

7)• Find the value of x.

5

x

7

3

Angle Measure and Segment Lengths (9-

7)• Find the value of y.

15

8

y

Angle Measure and Segment Lengths (9-

7)

a

8

8

b

12

34

x

90

30

O

Angle Measure and Segment Lengths (9-

7)

a 21

98

x7

8

M