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Slide 1 / 150

Geometry

Circles

www.njctl.org

2014-06-03

Slide 2 / 150

Table of Contents

Parts of a CircleAngles & ArcsChords, Inscribed Angles & Polygons

Segments & CirclesEquations of a Circle

Click on a topic to go to that section

Tangents & Secants

Area of a Sector

Slide 3 / 150

http://www.njctl.org





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Parts of a Circle

Return to the table of contents

Slide 4 / 150

A circle is the set of all points in a plane that are a fixed distance from a given point in the plane called the center.

center

Slide 5 / 150

The symbol for a circle is and is named by a capital letter placed by the center of the circle.

.

A

B

(circle A or . A)is a radius of . A

A radius (plural, radii) is a line segment drawn from the center of the circle to any point on the circle. It follows from the definition of a circle that all radii of a circle are congruent.

.is a radius of A).(circle A or

A

.

Slide 6 / 150

page1svg

A

M

C

R

T

is the diameter of circle A

is a chord of circleAA chord is a segment that has its endpoints on the circle.

A diameter is a chord that goes through the center of the circle. All diameters of a circle are congruent.

What are the radii in this diagram?

Slide 7 / 150

A

M

C

R

T

is the diameter of circle A

is a chord of circleAA chord is a segment that has its endpoints on the circle.

A diameter is a chord that goes through the center of the circle. All diameters of a circle are congruent.

What are the radii in this diagram?[This object is a pull tab]

Ans

wer

&

Slide 7 (Answer) / 150

The relationship between the diameter and the radius

A

The measure of the diameter, d, is twice the measure of the radius, r.

Therefore, orM

C

T

If then what is the length of ,

In . A

what is the length of

Slide 8 / 150

The relationship between the diameter and the radius

A

The measure of the diameter, d, is twice the measure of the radius, r.

Therefore, orM

C

T

If then what is the length of ,

In . A

what is the length of

[This object is a pull tab]

Ans

wer AC = 5

TC = 10


1 A diameter of a circle is the longest chord of the circle.True

False

Slide 9 / 150

1 A diameter of a circle is the longest chord of the circle.True

False


Ans

wer

True


2 A radius of a circle is a chord of a circle.

True

False

Slide 10 / 150

2 A radius of a circle is a chord of a circle.

True

False


Ans

wer

False


3 Two radii of a circle always equal the length of a diameter of a circle.

True

False

Slide 11 / 150

3 Two radii of a circle always equal the length of a diameter of a circle.

True

False


Ans

wer

True


4 If the radius of a circle measures 3.8 meters, what is the measure of the diameter?

Slide 12 / 150

4 If the radius of a circle measures 3.8 meters, what is the measure of the diameter?


Ans

wer

7.6 m


5 How many diameters can be drawn in a circle?

A 1

B 2

C 4

D infinitely many

Slide 13 / 150

5 How many diameters can be drawn in a circle?

A 1

B 2

C 4

D infinitely many


Ans

wer

D


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

B

D

E k

l

line l is a secant of this circle.

A tangent is a line in the plane of a circle that intersects the circle at exactly one point (the point of tangency).

line k is a tangent D is the point of tangency.

tangent ray, , and the tangent segment, , are also called tangents. They must be part of a tangent line.

Note: This is not a tangent ray.

Slide 14 / 150

COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points.

Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric.

2 points tangent circles

1 point

concentric circles

....

.

no points

Slide 15 / 150

A Common Tangent is a line, ray, or segment that is tangent to 2 coplanar circles.

Internally tangent(tangent line

passes between them)

Externally tangent(tangent line does not pass between

them)

Slide 16 / 150

6 How many common tangent lines do the circles have?

Slide 17 / 150



Ans

wer

4



Slide 18 / 150



Ans

wer

1



Slide 19 / 150



Ans

wer

2



Slide 20 / 150



Ans

wer

0


Using the diagram below, match the notation with the term that best describes it:

A

C

D

E

F

G.

.

..

. .

B.

centerradiuschord

diametersecanttangent point of tangency

common tangent

Slide 21 / 150

Using the diagram below, match the notation with the term that best describes it:

A

C

D

E

F

G.

.

..

. .

B.

centerradiuschord

diametersecanttangent point of tangency

common tangent[This object is a pull tab]

Ans

wer

CenterCommon TangentChordSecantTangentPoint of TangencyRadiusDiameter


Angles & Arcs


Slide 22 / 150

An ARC is an unbroken piece of a circle with endpoints on the circle.

..

A

B

Arc of the circle or AB

Arcs are measured in two ways:1) As the measure of the central angle in degrees2) As the length of the arc itself in linear units

(Recall that the measure of the whole circle is 360o.)

Slide 23 / 150

A central angle is an angle whose vertex is the center of the circle.

M

AT

HS. .In , is the central

angle.

Name another central angle.

Slide 24 / 150

page1svg

page1svg

A central angle is an angle whose vertex is the center of the circle.

M

AT

HS. .In , is the central

angle.

Name another central angle.[This object is a pull tab]

Ans

wer


M

AT

HS. .minor arc MA

If is less than 1800, then the points on that lie in the interior of form the minor arc with endpoints M and H.

Name another minor arc.

MAHighlight

Slide 25 / 150

M

AT

HS. .minor arc MA

If is less than 1800, then the points on that lie in the interior of form the minor arc with endpoints M and H.

Name another minor arc.

MAHighlight[This object is a pull tab]

Ans

wer


M

AT

HS. .major arc

Points M and A and all points of exterior to form a major arc, Major arcs are the "long way" around the circle.

Major arcs are greater than 180o. Highlight

Major arcs are named by their endpoints and a point on the arc.

Name another major arc.

MSA

MSA

Slide 26 / 150

M

AT

HS. .major arc

Points M and A and all points of exterior to form a major arc, Major arcs are the "long way" around the circle.

Major arcs are greater than 180o. Highlight

Major arcs are named by their endpoints and a point on the arc.

Name another major arc.

MSA

MSA


Ans

wer


M

AT

HS. . minor arc

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

MAT is a semicircle. Highlight the semicircle.

Semicircles are named by their endpoints and a point on the arc.

Name another semicircle.

Slide 27 / 150

M

AT

HS. . minor arc

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

MAT is a semicircle. Highlight the semicircle.

Semicircles are named by their endpoints and a point on the arc.

Name another semicircle.


Ans

wer


The measure of a minor arc is the measure of its central angle.The measure of the major arc is 3600 minus the measure of the central angle.

Measurement By A Central Angle

A

B

D.

400

G

400

3600 - 400 = 3200

Slide 28 / 150

The Length of the Arc Itself (AKA - Arc Length)

Arc length is a portion of the circumference of a circle.

Arc Length Corollary - In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 3600.

C

A

T

r

arc length of =3600

CT CT

CT CTarc length of =3600

.or

Slide 29 / 150

C

A

T

8 cm

600

EXAMPLE

In , the central angle is 600 and the radius is 8 cm.Find the length of

A

CT

Slide 30 / 150


EXAMPLE

S

A

Y

4.19 in

400

AIn , the central angle is 400 and the length of is 4.19 in. Find the circumference of A.

SY

A.In , the central angle is 400 and the length of is 4.19 in. Find the circumference of

SYA

Slide 31 / 150

EXAMPLE

S

A

Y

4.19 in

400

AIn , the central angle is 400 and the length of is 4.19 in. Find the circumference of A.

SY

A.In , the central angle is 400 and the length of is 4.19 in. Find the circumference of

SYA


Ans

wer

arc length of =3600

SY SY

=3600

4004.19

4.19 =91

= 37.71 in


10 In circle C where is a diameter, find

1350

A

C

B

D

15 in

Slide 32 / 150

10 In circle C where is a diameter, find

1350

A

C

B

D

15 in


Ans

wer


11 In circle C, where is a diameter, find

1350

A

C

B

D

15 in

Slide 33 / 150


1350

A

C

B

D

15 in


Ans

wer



1350

A

C

B

D

15 in

Slide 34 / 150


1350

A

C

B

D

15 in


Ans

wer


13 In circle C can it be assumed that AB is a diameter?

Yes

No 1350

A

C

B

D

Slide 35 / 150

13 In circle C can it be assumed that AB is a diameter?

Yes

No 1350

A

C

B

D


Ans

wer

Yes


14 Find the length of

450

A

C

3 cm

B

Slide 36 / 150

14 Find the length of

450

A

C

3 cm

B


Ans

wer


15 Find the circumference of circle T.

T

750

6.82 cm

Slide 37 / 150

15 Find the circumference of circle T.

T

750

6.82 cm


Ans

wer


1400

16 In circle T, WY & XZ are diameters. WY = XZ = 6.

If XY = , what is the length of YZ?

A

B

C

D

T

W

Y

X

Z

Slide 38 / 150

1400

16 In circle T, WY & XZ are diameters. WY = XZ = 6.

If XY = , what is the length of YZ?

A

B

C

D

T

W

Y

X

Z[This object is a pull tab]

Ans

wer

A


Adjacent arcs: two arcs of the same circle are adjacent if they have a common endpoint.

Just as with adjacent angles, measures of adjacent arcs can be added to find the measure of the arc formed by the adjacent arcs.

ADJACENT ARCS

.

..

C

A

T

+=

Slide 39 / 150

EXAMPLEA result of a survey about the ages of people in a city are shown. Find the indicated measures.

>65

45-64

15-17

17-44

S

U

V

R

300

900

800

600

1000

T

1.

2.

3.

4.

Slide 40 / 150

EXAMPLEA result of a survey about the ages of people in a city are shown. Find the indicated measures.

>65

45-64

15-17

17-44

S

U

V

R

300

900

800

600

1000

T

1.

2.

3.

4.


Ans

wer

= 600 + 800 = 1400

1000 + 300 = 1300

= 600 + 800 + 900 = 2300

= 3600 - 900 = 2700


Match the type of arc and it's measure to the given arcs below:

1200

800 600

T

SR

Q

minor arc major arc semicircle

1200 240018001600800

Slide 41 / 150

Match the type of arc and it's measure to the given arcs below:

1200

800 600

T

SR

Q

minor arc major arc semicircle

1200 240018001600800


Teac

her N

otes

Arc labels and measurements in the box are infinitely cloned so they can be pulled up and matched with the arc.


CONGRUENT CIRCLES & ARCS· Two circles are congruent if they have the same radius.· Two arcs are congruent if they have the same measure and they are arcs of the same circle or congruent circles.

C

D E

F550 550

R

S

T

U

& because they are in the same circle and

have the same measure, but are not congruent because they are arcs of circles that are not congruent.

Slide 42 / 150

17

True

False1800

700

400

A

B

C

D

Slide 43 / 150

17

True

False1800

700

400

A

B

C

D


Ans

wer True


18

True

False 850

M

N

L

P

Slide 44 / 150

18

True

False 850

M

N

L

P

[This object is a pull tab]A

nsw

er False


90019 Circle P has a radius of 3 and has a measure

of . What is the length of ?

A

B

C

D

P

A

B

Slide 45 / 150

90019 Circle P has a radius of 3 and has a measure

of . What is the length of ?

A

B

C

D

P

A

B


Ans

wer A


20 Two concentric circles always have congruent radii.

True

False

Slide 46 / 150

20 Two concentric circles always have congruent radii.

True

False


Ans

wer

False


21 If two circles have the same center, they are congruent.

True

False

Slide 47 / 150

21 If two circles have the same center, they are congruent.

True

False


Ans

wer

False


22 Tanny cuts a pie into 6 congruent pieces. What is the measure of the central angle of each piece?

Slide 48 / 150

22 Tanny cuts a pie into 6 congruent pieces. What is the measure of the central angle of each piece?


Ans

wer


Chords, Inscribed Angles & Polygons


Slide 49 / 150

Lab - Properties of Chords

Click on the link below and complete the labs before the Chords lesson.

Slide 50 / 150

is the arc of

When a minor arc and a chord have the same endpoints, we call the arc The Arc of the Chord.

.C

P

Q

**Recall the definition of a chord - a segment with endpoints on the circle.

Slide 51 / 150

page1svg

https://njctl.org/courses/math/geometry/circles/properties-of-chords-lab/

THEOREM:

In a circle, if one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

T

P

SQ

E is the perpendicular bisector of .

Therefore, is a diameter of the circle.

Likewise, the perpendicular bisector of a chord of a circle passes through the center of a circle.

Slide 52 / 150

THEOREM:If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

A

C

E

S

X.

is a diameter of the circle and is perpendicular to chord

Therefore,

Slide 53 / 150

THEOREM:In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

A

B C

D

iff

*iff stands for "if and only if"

Slide 54 / 150

If , then point Y and any line segment, or ray, that contains Y, bisects

BISECTING ARCS

C

X

Z

Y

Slide 55 / 150

Find:,, and

EXAMPLE

A

BC

D

E

. (9x)0

(80 - x)0

and, ,Find:

Slide 56 / 150

Find:,, and

EXAMPLE

A

BC

D

E

. (9x)0

(80 - x)0

and, ,Find:


Ans

wer

= 9(8) = 720

= 80 - 8 = 720

= 720 + 720 = 1440

9x = 80 - x10x = 80 x = 8


THEOREM:In the same circle, or congruent circles, two chords are congruent if and only if they are equidistant from the center.

.C

G

DEA

FB

iff

Slide 57 / 150

EXAMPLE

Given circle C, QR = ST = 16. Find CU.

.Q

R

S

T

U

V

2x

5x - 9C

Since the chords QR & ST are congruent, they are equidistant from C. Therefore,

Slide 58 / 150

EXAMPLE

Given circle C, QR = ST = 16. Find CU.

.Q

R

S

T

U

V

2x

5x - 9C

Since the chords QR & ST are congruent, they are equidistant from C. Therefore,


Ans

wer

2x = 5x - 99 = 3x CU = 2(3) = 63 = x


23 In circle R, and . FindA

B

C

D

R.1080

Slide 59 / 150

23 In circle R, and . FindA

B

C

D

R.1080


Ans

wer 1080


24 Given circle C below, the length of is:

A 5

B 10

C 15

D 20

D B

F

C.10

A

Slide 60 / 150

24 Given circle C below, the length of is:

A 5

B 10

C 15

D 20

D B

F

C.10

A


Ans

wer D


25 Given: circle P, PV = PW, QR = 2x + 6, and ST = 3x - 1. Find the length of QR.

A 1

B 7C 20

D 8

R

SQ

T

P

W

.V

Slide 61 / 150

25 Given: circle P, PV = PW, QR = 2x + 6, and ST = 3x - 1. Find the length of QR.

A 1

B 7C 20

D 8

R

SQ

T

P

W

.V


Ans

wer

C


26 AH is a diameter of the circle.

True

False

A

S

H

M

3

3

5

T

Slide 62 / 150

26 AH is a diameter of the circle.

True

False

A

S

H

M

3

3

5

T


Ans

wer

False


INSCRIBED ANGLESD

OG

Inscribed angles are angles whose vertices are in on the circle and whose sides are chords of the circle.

The arc that lies in the interior of an inscribed angle, and has endpoints on the angle, is called the intercepted arc.

is an inscribed angle and is its intercepted arc.

Lab - Inscribed Angles

Click on the link below and complete the lab.

Slide 63 / 150

https://njctl.org/courses/math/geometry/circles/inscribed-angles-lab/

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

C

A

T

Slide 64 / 150

Slide 65 / 150


THEOREM:If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

D

C

B

Asince they both intercept

Slide 66 / 150

In a circle, parallel chords intercept congruent arcs.

O

B

.A

DC

In circle O, if , then, thenIn circle O, if

Slide 67 / 150

27 Given circle C below, find

D E

C

A B

. 1000

350

Slide 68 / 150


D E

C

A B

. 1000

350


Ans

wer 500



D E

C

A B

. 1000

350

Slide 69 / 150


D E

C

A B

. 1000

350


Ans

wer 1100


29 Given the figure below, which pairs of angles are congruent?

A

B

C

D

R

S

U

T

Slide 70 / 150

29 Given the figure below, which pairs of angles are congruent?

A

B

C

D

R

S

U

T


Ans

wer A


Slide 71 / 150


31 In a circle, two parallel chords on opposite sides of the center have arcs which measure 1000 and 1200. Find the measure of one of the arcs included between the chords.

Slide 72 / 150

31 In a circle, two parallel chords on opposite sides of the center have arcs which measure 1000 and 1200. Find the measure of one of the arcs included between the chords.


Ans

wer 700


32 Given circle O, find the value of x.

.O

A B

C D

x

300

Slide 73 / 150


.O

A B

C D

x

300


Ans

wer 1200



.O

A B

C D

x

1000

350

Slide 74 / 150


.O

A B

C D

x

1000

350


Ans

wer 1200


In the circle below, and Find

, and

Try This

P

S

1

2

3

4

Q

T

Slide 75 / 150

In the circle below, and Find

, and

Try This

P

S

1

2

3

4

Q

T


Ans

wer


INSCRIBED POLYGONS

A polygon is inscribed if all its vertices lie on a circle.

.

.

.

inscribed triangle

.

.

.

.

inscribed quadrilateral

Slide 76 / 150

THEOREM:If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

A

L

G

x.iff AC is a diameter of the circle.

Slide 77 / 150

Slide 78 / 150

EXAMPLE

Find the value of each variable:

2a

2a4b

2bL

K

J

M

Slide 79 / 150

EXAMPLE

Find the value of each variable:

2a

2a4b

2bL

K

J

M


Ans

wer 2a + 2a = 180

4a = 180 a = 450

4b + 2b = 180 6b = 180 b = 300


34 The value of x is

A

B

C

D

1500

980

1120

1800

C

B

A

Dx

y

680

820

Slide 80 / 150

34 The value of x is

A

B

C

D

1500

980

1120

1800

C

B

A

Dx

y

680

820


Ans

wer

B


35 In the diagram, is a central angle and . What is ?

150

300

600

1200

A

B

C

D

.B

A

DC

1200

600

300

150

Slide 81 / 150

35 In the diagram, is a central angle and . What is ?

150

300

600

1200

A

B

C

D

.B

A

DC

1200

600

300

150


Ans

wer B


36 What is the value of x?

A 5

B 10

C 13

D 15

E

F G

(12x + 40)0

(8x + 10)0

Slide 82 / 150


Tangents & Secants


Slide 83 / 150

page1svg

**Recall the definition of a tangent line: A line that intersects the circle in exactly one point.

THEOREM:In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle (the point of tangency).

..X

B

l

lLine is tangent to circle X iff would be the point of tangency.BBLine is tangent to circle X iff would be the point of tangency.

l l

Lab - Tangent Lines

Click on the link below and complete the lab.

Slide 84 / 150

Verify A Line is Tangent to a Circle

.T

P

S

35

37

12}

Given: is a radius of circle PIs tangent to circle P?

Slide 85 / 150

Verify A Line is Tangent to a Circle

.T

P

S

35

37

12}Given: is a radius of circle P

Is tangent to circle P?


Ans

wer

Since 352 + 122 = 372, triangle PST is a right triangle. Therefore, ST is perpendicular to radius TP at its endpoint on circle P. So, ST is tangent to circle P at T.


https://njctl.org/courses/math/geometry/circles/tangent-lines/

Finding the Radius of a Circle

.A

C

B

r

r

50 ft

80 ft

If B is a point of tangency, find the radius of circle C.

Slide 86 / 150

Finding the Radius of a Circle

.A

C

B

r

r

50 ft

80 ft

If B is a point of tangency, find the radius of circle C.


Ans

wer

AC2 + BC2 = AB2

802 + r2 = (50 + r)2 6400 + r2 = r2 + 100r + 2500 6400 = 100r + 2500 3900 = 100r 39 = rSo, r = 39 ft.


THEOREM:Tangent segments from a common external point are congruent.

R

A

T

P.

If AR and AT are tangent segments, then

Slide 87 / 150

EXAMPLE

Given: RS is tangent to circle C at S and RT is tangent to circle C at T. Find x.

S

R

T

C.28

3x + 4

Slide 88 / 150

EXAMPLE

Given: RS is tangent to circle C at S and RT is tangent to circle C at T. Find x.

S

R

T

C.28

3x + 4


Ans

wer 3x + 4 = 28

3x = 24 x = 8


37 AB is a radius of circle A. Is BC tangent to circle A?

Yes

No

.

B C

A

60

25

67

}

Slide 89 / 150

37 AB is a radius of circle A. Is BC tangent to circle A?

Yes

No

.

B C

A

60

25

67

}


Ans

wer

No


38 S is a point of tangency. Find the radius r of circle T.

A 31.7

B 60

C 14

D 3.5

.T

SR

r

r

48 cm

36 cm

Slide 90 / 150

38 S is a point of tangency. Find the radius r of circle T.

A 31.7

B 60

C 14

D 3.5

.T

SR

r

r

48 cm

36 cm


Ans

wer

C


39 In circle C, DA is tangent at A and DB is tangent at B. Find x.

A

D

B

C.25

3x - 8

Slide 91 / 150

39 In circle C, DA is tangent at A and DB is tangent at B. Find x.

A

D

B

C.25

3x - 8


Ans

wer


40 AB, BC, and CA are tangents to circle O. AD = 5, AC= 8, and BE = 4. Find the perimeter of triangle ABC.

.

B

E

F

AC D

O

Slide 92 / 150

40 AB, BC, and CA are tangents to circle O. AD = 5, AC= 8, and BE = 4. Find the perimeter of triangle ABC.

.

B

E

F

AC D

O


Ans

wer


Tangents and secants can form other angle relationships in circle. Recall the measure of an inscribed angle is 1/2 its intercepted arc. This can be extended to any angle that has its vertex on the circle. This includes angles formed by two secants, a secant and a tangent, a tangent and a chord, and two tangents.

Slide 93 / 150

A Tangent and a Chord

THEOREM:If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

..

.

A

M

R

2 1

Slide 94 / 150

Slide 95 / 150

THEOREM:If two chords intersect inside a circle, then the measure of each angle is half the sum of the intercepted arcs by the angle and vertical angle.

M A

H T

12

Slide 96 / 150

EXAMPLE

Find the value of x.

D

C

A

B

x0 7601780

Slide 97 / 150

EXAMPLE


D

C

A

B

x0 7601780


Ans

wer


EXAMPLE

Find the value of x.1300

x0

1560

Slide 98 / 150

EXAMPLE

Find the value of x.1300

x0

1560[This object is a pull tab]

Ans

wer x = 1/2 (1300 + 1560)

x = 1430


41 Find the value of x.

C

H

DFx0

780

420

E

Slide 99 / 150


C

H

DFx0

780

420

E


Ans

wer



340

(x + 6)0

(3x - 2)0

Slide 100 / 150


340

(x + 6)0

(3x - 2)0


Ans

wer


43 Find

A

B

650

Slide 101 / 150

43 Find

A

B

650[This object is a pull tab]

Ans

wer


44 Find

12600

Slide 102 / 150

44 Find

12600


Ans

wer



x122.50

450

Slide 103 / 150


x122.50

450


Ans

wer


2470

A

B

x0

To find the angle, you need the measure of both intercepted arcs. First, find the measure of the minor arc . Then we can calculate the measure of the angle . x0

Slide 104 / 150

2470

A

B

x0

To find the angle, you need the measure of both intercepted arcs. First, find the measure of the minor arc . Then we can calculate the measure of the angle . x0


Ans

wer

First find the minor arc.



Students type their answers here

2200

x0

Slide 105 / 150


Students type their answers here

2200

x0


Ans

wer

First find the minor arc.


47 Find the value of x. Students type their answers here

x01000

Slide 106 / 150


x01000


Ans

wer

First find the major arc.


48 Find the value of x Students type their answers here

x0

500

Slide 107 / 150

48 Find the value of x Students type their answers here

x0

500


Ans

wer

Find the major arc.



1200

(5x + 10)0

Slide 108 / 150


1200

(5x + 10)0


Ans

wer

Find the major arc.



(2x - 30)0

300 x

Slide 109 / 150


Segments & Circles


Slide 110 / 150

THEOREM:If two chords intersect inside a circle, then the products of the measures of the segments of the chords are equal.

AC

D B

E

Slide 111 / 150

page1svg

EXAMPLE


5

5

x

4

Slide 112 / 150

EXAMPLE


5

5

x

4


Ans

wer


EXAMPLEFind ML & JK.

x + 2x +

4

x x + 1

M K

J

L

Slide 113 / 150

EXAMPLEFind ML & JK.

x + 2x +

4

x x + 1

M K

J

L


Ans

wer

ML = (2 + 2) +( 2 + 1) = 7

JK = 2 + (2 + 4) = 8



189

16

x

Slide 114 / 150


189

16

x


Ans

wer



A -2

B 4

C 5

D 6 x2

2x + 6

x

Slide 115 / 150


A -2

B 4

C 5

D 6 x2

2x + 6

x


Ans

wer

D


THEOREM:If two secant segments are drawn to a circle from an external point, then the product of the measures of one secant segment and its external secant segment equals the product of the measures of the other secant segment and its external secant segment.

A

B

EC D

Slide 116 / 150

EXAMPLE


9 6

x 5

Slide 117 / 150

EXAMPLE


9 6

x 5


Ans

wer



3

x + 2x + 1

x - 1

Slide 118 / 150


3

x + 2x + 1

x - 1


Ans

wer



x + 4

x - 2

5

4

Slide 119 / 150


x + 4

x - 2

5

4


Ans

wer


THEOREM:If a tangent segment and a secant segment are drawn to a circle from an external point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.

A

E CD

Slide 120 / 150

EXAMPLEFind RS.

R S

Q

T

16

x 8

Slide 121 / 150



1

x

3

Slide 122 / 150


1

x

3


Ans

wer



x12

24

Slide 123 / 150


x12

24


Ans

wer


Equations of a Circle


Slide 124 / 150

y

x

r

(x, y)Let (x, y) be any point on a circle with center at the origin and radius, r. By the Pythagorean Theorem,

x2 + y2 = r2

This is the equation of a circle with center at the origin.

Slide 125 / 150

page1svg

EXAMPLE

Write the equation of the circle.

4

Slide 126 / 150

EXAMPLE

Write the equation of the circle.

4


Ans

wer x2 + y2 = (4)2

x2 + y2 = 16


For circles whose center is not at the origin, we use the distance formula to derive the equation.

.(x, y)

(h, k)

r

This is the standard equation of a circle.

Slide 127 / 150

EXAMPLE

Write the standard equation of a circle with center (-2, 3) & radius 3.8.

Slide 128 / 150

EXAMPLE

Write the standard equation of a circle with center (-2, 3) & radius 3.8.


Ans

wer


EXAMPLE

The point (-5, 6) is on a circle with center (-1, 3). Write the standard equation of the circle.

Slide 129 / 150


EXAMPLE

The equation of a circle is (x - 4)2 + (y + 2)2 = 36. Graph the circle.

We know the center of the circle is (4, -2) and the radius is 6.

..

..

First plot the center and move 6 places in each direction.

Then draw the circle.

Slide 130 / 150

57 What is the standard equation of the circle below?

A

B

C

D

x2 + y2 = 400

(x - 10)2 + (y - 10)2 = 400

(x + 10)2 + (y - 10)2 = 400

(x - 10)2 + (y + 10)2 = 40010

Slide 131 / 150

57 What is the standard equation of the circle below?

A

B

C

D

x2 + y2 = 400

(x - 10)2 + (y - 10)2 = 400

(x + 10)2 + (y - 10)2 = 400

(x - 10)2 + (y + 10)2 = 40010

[This object is a pull tab]A

nsw

er A


58 What is the standard equation of the circle?A

B

C

D

(x - 4)2 + (y - 3)2 = 9

(x + 4)2 + (y + 3)2 = 9

(x + 4)2 + (y + 3)2 = 81

(x - 4)2 + (y - 3)2 = 81

Slide 132 / 150

58 What is the standard equation of the circle?A

B

C

D

(x - 4)2 + (y - 3)2 = 9

(x + 4)2 + (y + 3)2 = 9

(x + 4)2 + (y + 3)2 = 81

(x - 4)2 + (y - 3)2 = 81


Ans

wer D


59 What is the center of (x - 4)2 + (y - 2)2 = 64?

A (0,0)

B (4,2)

C (-4, -2)

D (4, -2)

Slide 133 / 150

59 What is the center of (x - 4)2 + (y - 2)2 = 64?

A (0,0)

B (4,2)

C (-4, -2)

D (4, -2)


Ans

wer B


60 What is the radius of (x - 4)2 + (y - 2)2 = 64?

Slide 134 / 150

60 What is the radius of (x - 4)2 + (y - 2)2 = 64?


Ans

wer

r=8


61 The standard equation of a circle is (x - 2)2 + (y + 1)2 = 16. What is the diameter of the circle?

A 2

B 4

C 8

D 16

Slide 135 / 150

61 The standard equation of a circle is (x - 2)2 + (y + 1)2 = 16. What is the diameter of the circle?

A 2

B 4

C 8

D 16


Ans

wer

C


62 Which point does not lie on the circle described by the equation (x + 2)2 + (y - 4)2 = 25?

A (-2, -1)

B (1, 8)

C (3, 4)

D (0, 5)

Slide 136 / 150

62 Which point does not lie on the circle described by the equation (x + 2)2 + (y - 4)2 = 25?

A (-2, -1)

B (1, 8)

C (3, 4)

D (0, 5)[This object is a pull tab]

Ans

wer D



Area of a Sector

Slide 137 / 150

page1svg

A sector of a circle is the portion of the circle enclosed by two radii and the arc that connects them.

A

B

C

Minor Sector

Major Sector

Slide 138 / 150

63 Which arc borders the minor sector?

A

B A

BC

D

Slide 139 / 150

63 Which arc borders the minor sector?

A

B A

BC

D


Ans

wer A


64 Which arc borders the major sector?

A

B

W

X

YZ

Slide 140 / 150

64 Which arc borders the major sector?

A

B

W

X

YZ


Ans

wer B


Lets think about the formula...The area of a circle is found by

We want to find the area of part of the circle, so the formula for the area of a sector is the fraction of the circle multiplied by the area of the circle

When the central angle is in degrees, the fraction of the circle is out of the total 360 degrees.

Slide 141 / 150

Finding the Area of a Sector1. Use the formula: when θ is in degrees

450

AB

C

r=3

Slide 142 / 150

Finding the Area of a Sector1. Use the formula: when θ is in degrees

450

AB

C

r=3


Ans

wer


Example:Find the Area of the major sector.

C

A

T

8 cm

600

Slide 143 / 150

Example:Find the Area of the major sector.

C

A

T

8 cm

600


Ans

wer

cm2


65 Find the area of the minor sector of the circle. Round your answer to the nearest hundredth.

C

A

T5.5 cm 300

Slide 144 / 150

65 Find the area of the minor sector of the circle. Round your answer to the nearest hundredth.

C

A

T5.5 cm 300


Ans

wer

cm2


66 Find the Area of the major sector for the circle. Round your answer to the nearest thousandth.

C

A

T

12 cm

850

Slide 145 / 150

66 Find the Area of the major sector for the circle. Round your answer to the nearest thousandth.

C

A

T

12 cm

850


Ans

wer

cm2


67 What is the central angle for the major sector of the circle?

C

A

G

15 cm

1200

Slide 146 / 150

67 What is the central angle for the major sector of the circle?

C

A

G

15 cm

1200


Ans

wer


68 Find the area of the major sector. Round to the nearest hundredth.

C

A

G

15 cm

1200

Slide 147 / 150

68 Find the area of the major sector. Round to the nearest hundredth.

C

A

G

15 cm

1200


Ans

wer

cm2


69 The sum of the major and minor sectors' areas is equal to the total area of the circle.

True

False

Slide 148 / 150

69 The sum of the major and minor sectors' areas is equal to the total area of the circle.

True

False


Ans

wer True


70 A group of 10 students orders pizza. They order 5 12" pizzas, that contain 8 slices each. If they split the pizzas equally, how many square inches of pizza does each student get?

Slide 149 / 150

70 A group of 10 students orders pizza. They order 5 12" pizzas, that contain 8 slices each. If they split the pizzas equally, how many square inches of pizza does each student get?


Ans

wer

Each student gets 4 pieces


71 You have a circular sprinkler in your yard. The sprinkler has a radius of 25 ft. How many square feet does the sprinkler water if it only rotates 120 degrees?

Slide 150 / 150

71 You have a circular sprinkler in your yard. The sprinkler has a radius of 25 ft. How many square feet does the sprinkler water if it only rotates 120 degrees?


Ans

wer


Slide 2 / 150 Geometry Circlescontent.njctl.org/.../circles-2-2014-06-03-3-slides-per-page-w-answers.… · Slide 1 / 150 Geometry Circles 2014-06-03 Slide 2 / 150 Table of Contents

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