Lesson 10.1Parts of a Circle

Today, we are going to…> identify segments and lines related

to circles> use properties of tangents to a circle

C

Circle C

Diameter = _ radius

C

A chord is

YX

AB

A

B

X

YN

BN

C

A secant is

A

B

X

Y

YX

AB

C

A tangent is

ABA

B

Y X

XY

internal tangents

Common Tangent Lines

external tangents

Common Tangent Lines

Two circles can intersect in 2, 1, or 0 points.

Draw 2 circles that have2 points of intersection

internally tangent circles

Draw two circles that have1 point of intersection

externally tangent circles

Draw two circles that have1 point of intersection

concentric circles

Draw two circles that have no point of intersection

9. What are the center and radius of circle A?

Center: Radius =

10. What are the center and radius of circle B?

Center: Radius =

11. Identify the intersection of the two circles.

12. Identify all common tangents of the two circles.

mABC =

A

B

C

Theorem 10.1 & 10.2A line is tangent to a circle if

and only if it is _____________ to the radius from the point of

tangency.

A

B

C

7

13. Find CA.

15D

C

B

AWhat is DA?

7

14. Find x.

15

x

6

C

B

A

xx

168

What is CA?

7

156

C

B

A

2610

24

How do we test if 3 segments create a right triangle?

15. Is AB a tangent?

7

156

C

B

A

178

12

16. Is AB a tangent?

17. Find the slope of line t.

A

C

A (3,0) and C (5, -1)

Slope of AC?

Slope of line t?

t

C

A tangent segment

A B

One endpoint is the point of tangency.

Theorem 10.3If 2 segments from the same

point outside a circle are tangent to the circle, then

they are congruent.

7x - 2

3x + 8

18. Find x.

A

C

B

x2 + 25

50

19. Find x.

A

C

B

Lesson 10.2Arcs and Chords

Today, we are going to…> use properties of arcs and chords

of circles

C

An angle whose vertex is the center of a circle is a

central angle.

A

B

C

Minor Arc - Major Arc

A

B

D

Minor Arc

AB

Major Arc

ADB

C A

B

D

60˚

m AB =

Measures of Arcs

C

Semicircle

m AED = m ABD = m AD

A

B

D

E

Find the measures of the arcs.

1. m BD

2. m DE

3. m FC

4. m BFD

D

E

F

B

C

100˚52˚

68˚

53˚?

AD and EB are diameters.

F

A

B

D

E

C

5. Find x, y, and z.

30˚

z˚

x˚

y˚

x =

y =

z =

Theorem 10.4

Two arcs are congruent if and only if their chords

are congruent.

(2x + 48)°(3x + 11)°

B

ADC

6. Find m AB

Theorem 10.5 & 10.6

A chord is a diameter if and only if it is a

perpendicular bisector of a chord and bisects its arc.

7. Is AB a diameter?A

B

8. Is AB a diameter?A

B

8

8

9. Is AB a diameter?A

B

Theorem 10.7

Two chords are congruent if and only if they are equidistant from the

center.

AB = 12

10. Find CG.

DE = 12

7D

G BA

C

F

E6

x

?

Lesson 10.3Inscribed Angles

Today, we are ALSO going to…> use properties of inscribed angles

to solve problems

An inscribed angle is an angle whose vertex is on the

circle and whose sides contain chords of the circle.

Theorem 10.8If an angle is inscribed,

then its measure is half the measure of its intercepted

arc.

x2x

1. Find x.

x°

120°

x = 60°

2. Find x.

x°

70°

x = 140°

Theorem 10.9If 2 inscribed angles

intercept the same arc, then the angles are

congruent.

3. Find x and y.

y°

45°

x°

InscribedPentagon

x°

A

D

C

B

4. DC is a diameter. Find x.

Theorem 10.10If a right triangle is inscribed in a circle, then the hypotenuse is a

diameter of the circle.

5. Find the values of x and y.

x°

y°A

42 D

C

B

Theorem 10.11If a quadrilateral is inscribed in a

circle, then its opposite angles are

supplementary.

21

4 3

m 1 + m 3 = 180º

m 2 + m 4 = 180º

6. Find the values of x and y.

x°

110°

80° y°

7. Find the values of x and y.

x°

120°

100° y°

Lesson 10.4Angle Relationships

in CirclesToday, we are going to…> use angles formed by tangents and

chords to solve problems > use angles formed by intersecting

lines to solve problems

Theorem 10.12

If a tangent and a chord intersect at a point on a

circle, then...

GSP

Theorem 10.12

… the measure of each angle formed is half the measure of its

intercepted arc.

1A

BC

2

1A

BC

2

1. Find m 1 and m 2.

100°

2. Find and mACB and mAB

95°A

B

C

3. Find x

5x°A

B

C(9x + 20)˚

Theorem 10.13If 2 chords intersect inside a circle, then…

A

B

C

D

1

B

CA

D

1

…the measure of the angle is half the sum of the intercepted arcs.

A

B

C

D

x°

4. Find x.100°

120°

A

B

C

D

x°

5. Find x.130°

160°

A

B

C

D

x°

6. Find x.

80° 90°y°

A

B

C

D

x°7. Find x.

100°

120°

A

B

C

D

x°

8. Find x.

52°74°

Do you notice a pattern?

Theorem 10.14If a tangent and a secant, two tangents, or two secants intersect outside a circle, then…

A

C

D

1

Theorem 10.14If a tangent and a secant, two tangents, or two secants intersect outside a circle, then…

A

B

C 1

Theorem 10.14If a tangent and a secant, two tangents, or two secants intersect outside a circle, then…

A

BC

D

1

A

BC

D1

…the measure of the angle is half the difference of the intercepted arcs.

9. Find x.

20° 80°

A

BC

D

x°

10. Find x.

24°90°

A

BC

Dx°

11. Find x.

200°x°

A

C

D

12. Find x.

135°x°

13. Find x.

100°

3 100°2 1

100°

60°

Lesson 10.5Segment Lengths

in CirclesToday, we are going to…> find the lengths of segments of chords, tangents, and secants

Theorem 10.15

If 2 chords intersect inside a circle, then the product of their “segments” are

equal.

a · b = c · d

a

b

c d

1. Find x.

6

8 4

x

2. Find x.

3x

182x

3

3. Find x.

2x

18x

4

Theorem 10.16 If 2 secant segments share the same endpoint outside

a circle, then…

GSP

…one secant segment times its external part

equals the other secant segment times its external part.

a · c = b · d

b

a

c

d

3. Find x.

5

x

4 6

4. Find x.

9

10x

20

Theorem 10.17 If a secant segment and a tangent segment share an endpoint outside a circle,

then…

…the length of the tangent segment squared equals the

length of the secant segment times its external

part.

a · a = b · d

db

a

a2 = b · d

54

x5. Find x.

15x

106. Find x.

Quadratic Formula?♫♪♫♪♫♪♫♪♫♪♫♪

15x

106. Find x.

x20

317. Find x.

8. Find x.

3

48

x

10x

89. Find x.

Lesson 10.6Equations of

CirclesToday, we are going to…> write the equation of a circle

Standard Equation for a Circle with

Center: (0,0) Radius = r

1. Write an equation of the circle.

2. Write an equation of the circle.

Standard Equation for a Circle with

Center: (h,k) Radius = r

3.Write an equation of the circle.

C =

r =

4.Write an equation of the circle.

C = r =

Graph (x – 3)2 + (y + 2)2 = 9

Center?

Radius =

Identify the center and radius of the circle with the given equation.

5. (x – 1)2 + (y + 3)2 = 100

6. x2 + (y - 7)2 = 8

7. (x + 1)2 + y2 = ¼

Center: (1, -3) radius = 10

Center: (0, 7) radius ≈ 2.83

Center: (-1, 0) radius = ½

Write the standard equation of the circle with a center of (5, -1) if a point on the circle is (1,2).

8. Write the standard equation of the circle with a center of (-3, 4) if a point on the circle is (2,-5).

Is (-2,-10) on the circle (x + 5)2 + (y + 6)2 = 25?

9. Is (0, - 6) on the circle (x + 5)2 + (y – 5)2 = 169?

10. Is (2, 5) on the circle (x – 7)2 + (y + 5)2 = 121?

><

=

Would the point be inside the circle, outside the circle, or on the circle?

(x – 13)2 + (y - 4)2 = 100

11. (11, 13)

12. (6, -5)

13. (19, - 4)

Lessons 11.4 & 11.5Circumference and

Area of CirclesToday, we are going to…> find the length around part of a circle and find the area of part of a circle

Circumference

Arc Length

=

A

B

A

B50°7 cm

1. Find the length of AB

A

B

85°

10 cm

2. Find the radius

3. Find the circumference.

Area

Sector of a circle A region bound by two radii &

their intercepted arc.

A slice of pizza!

Area of a Sector

=

3. Find the area of the sector.A

B50°7 cm

4. Find the radius. A

B

100°

3. Find the area.

Workbook

P. 211 (1 – 10)

P. 215 (1 – 6)