Geometry Circles - Currituck County Schools

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Slide 1 / 149

Geometry

Circles

www.njctl.org

2014-03-31

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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 / 149

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Parts of a Circle

Return to the table of contents

Slide 4 / 149

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 / 149

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 / 149

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?

Ans

wer

Slide 7 / 149

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) / 149

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

Ans

wer

Slide 8 / 149

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

Ans

wer

Slide 9 / 149

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

Ans

wer

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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

Ans

wer

Slide 11 / 149

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?

Ans

wer

Slide 12 / 149

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 4D infinitely many

Ans

wer

Slide 13 / 149

5 How many diameters can be drawn in a circle?

A 1

B 2

C 4D 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 / 149

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 / 149

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 / 149

6 How many common tangent lines do the circles have?

Ans

wer

Slide 17 / 149



Ans

wer

4



Ans

wer

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Ans

wer

1



Ans

wer

Slide 19 / 149



Ans

wer

2



Ans

wer

Slide 20 / 149



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

Ans

wer

Slide 21 / 149

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


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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 / 149

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.

Ans

wer

Slide 24 / 149

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

Ans

wer

Slide 25 / 149

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

Ans

wer

Slide 26 / 149

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.

Ans

wer

Slide 27 / 149

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 / 149

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 / 149

C

A

T

8 cm

600

EXAMPLE

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

ACT

Ans

wer

Slide 30 / 149

C

A

T

8 cm

600

EXAMPLE

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

ACT


Ans

wer

CT CTarc length of =3600

.= 600

3600.

8.38 cm


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.

SYA.

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

SYA

Ans

wer

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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.

SYA.

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

=36004004.19

4.19 =91

= 37.71 in


10 In circle C where is a diameter, find

1350

A

C

B

D

15 in

Ans

wer

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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

Ans

wer

Slide 33 / 149


1350

A

C

B

D

15 in


Ans

wer



1350

A

C

B

D

15 in

Ans

wer

Slide 34 / 149


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 Ans

wer

Slide 35 / 149

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

C3 cm

B

Ans

wer

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14 Find the length of

450

A

C3 cm

B


Ans

wer


15 Find the circumference of circle T.

T750

6.82 cm

Ans

wer

Slide 37 / 149

15 Find the circumference of circle T.

T750

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

Ans

wer

Slide 38 / 149

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 / 149

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

Slide 40 / 149

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

Teac

her N

otes

Slide 41 / 149

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 / 149

17

True

False1800

700

400

A

B

C

D

Ans

wer

Slide 43 / 149

17

True

False1800

700

400

A

B

C

D


Ans

wer True


18

True

False 850

M

N

L

P

Ans

wer

Slide 44 / 149

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

Ans

wer

Slide 45 / 149

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

Ans

wer

Slide 46 / 149

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

Ans

wer

Slide 47 / 149

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?

Ans

wer

Slide 48 / 149

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 / 149

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 50 / 149

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 51 / 149

page1svg

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

AC

E

SX.

is a diameter of the circle and is perpendicular to chord

Therefore,

Slide 52 / 149

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

if f

*iff stands for "if and only if"

Slide 53 / 149

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

BISECTING ARCS

C

X

Z

Y

Slide 54 / 149

Find:,, and

EXAMPLE

A

BC

D

E

. (9x)0

(80 - x)0

and, ,Find:

Ans

wer

Slide 55 / 149

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 56 / 149

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

Slide 57 / 149

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

Ans

wer

Slide 58 / 149

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

Ans

wer

Slide 59 / 149

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

Ans

wer

Slide 60 / 149

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

Ans

wer

Slide 61 / 149

26 AH is a diameter of the circle.

True

False

A

S

H

M

3

3

5

T

[This object is a pull tab]A

nsw

er

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.

Slide 62 / 149

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

C

A

T

Slide 63 / 149

EXAMPLE

Q R

T S

P.500

480

Find and

Ans

wer

Slide 64 / 149

EXAMPLE

Q R

T S

P.500

480

Find and


Ans

wer


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 65 / 149

In a circle, parallel chords intercept congruent arcs.

O

B

.A

DC

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

Slide 66 / 149

27 Given circle C below, find

D E

C

A B

. 1000

350 Ans

wer

Slide 67 / 149


D E

C

A B

. 1000

350


Ans

wer 500



D E

C

A B

. 1000

350

Ans

wer

Slide 68 / 149


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

RS

U

T

Ans

wer

Slide 69 / 149

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

A

B

C

D

RS

U

T


Ans

wer A


30 Find

X

Y

Z

P.

Ans

wer

Slide 70 / 149

30 Find

X

Y

Z

P.


Ans

wer 900


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

Slide 71 / 149

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

Ans

wer

Slide 72 / 149


.O

A B

C D

x

300


Ans

wer 1200



.O

A B

C D

x

1000

350

Ans

wer

Slide 73 / 149


.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

Ans

wer

Slide 74 / 149

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 75 / 149

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 76 / 149

THEOREM:A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

N

E

R

AC. N, E, A, and R lie on circle C iff

Slide 77 / 149

EXAMPLE

Find the value of each variable:

2a

2a4b

2bL

K

J

M

Ans

wer

Slide 78 / 149

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

Ans

wer

Slide 79 / 149

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

Ans

wer

Slide 80 / 149

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

Ans

wer

Slide 81 / 149

36 What is the value of x?

A 5

B 10

C 13

D 15

E

F G

(12x + 40)0

(8x + 10)0


Ans

wer

A


Tangents & Secants


Slide 82 / 149

**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

Slide 83 / 149

page1svg

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?

Ans

wer

Slide 84 / 149

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?


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.


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

Slide 85 / 149

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 86 / 149

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

Slide 87 / 149

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

}

Ans

wer

Slide 88 / 149

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

Ans

wer

Slide 89 / 149

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

Ans

wer

Slide 90 / 149

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

EF

AC D

O

Ans

wer

Slide 91 / 149

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

.

B

EF

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 92 / 149

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

R2 1

Slide 93 / 149

A Tangent and a Secant, Two Tangents, and Two Secants

THEOREM:If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is half the difference of its intercepted arcs.

A

B

C

1

a tangent and a secant

PQ

M

2

.

two tangents two secants

W

X

YZ

3

Slide 94 / 149

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 95 / 149

EXAMPLE

Find the value of x.

D

C

AB

x0 7601780

Ans

wer

Slide 96 / 149

EXAMPLE


D

C

AB

x0 7601780


Ans

wer


EXAMPLE

Find the value of x.1300

x0

1560

Ans

wer

Slide 97 / 149

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

Ans

wer

Slide 98 / 149


C

H

DFx0

780

420

E


Ans

wer



340

(x + 6)0

(3x - 2)0

Ans

wer

Slide 99 / 149


340

(x + 6)0

(3x - 2)0


Ans

wer


43 Find

A

B

650

Ans

wer

Slide 100 / 149

43 Find

A

B

650[This object is a pull tab]

Ans

wer


44 Find

12600

Ans

wer

Slide 101 / 149

44 Find

12600


Ans

wer



x122.50

450

Ans

wer

Slide 102 / 149


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

Ans

wer

Slide 103 / 149

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

Ans

wer

Slide 104 / 149


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 A

nsw

er

Slide 105 / 149


x01000


Ans

wer

First find the major arc.


48 Find the value of x Students type their answers here

x0

500

Ans

wer

Slide 106 / 149

48 Find the value of x Students type their answers here

x0

500


Ans

wer

Find the major arc.



1200

(5x + 10)0

Ans

wer

Slide 107 / 149


1200

(5x + 10)0


Ans

wer

Find the major arc.



(2x - 30)0

300 x

Ans

wer

Slide 108 / 149


(2x - 30)0

300 x[This object is a pull tab]

Ans

wer


Segments & Circles


Slide 109 / 149

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

A C

D B

E

Slide 110 / 149

EXAMPLE


5

5

x

4

Ans

wer

Slide 111 / 149

page1svg

EXAMPLE


5

5

x

4


Ans

wer


EXAMPLEFind ML & JK.

x + 2x +

4

x x + 1

M K

J

L

Ans

wer

Slide 112 / 149

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

Ans

wer

Slide 113 / 149


189

16

x


Ans

wer



A -2

B 4

C 5

D 6 x2

2x + 6

x

Ans

wer

Slide 114 / 149


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

E C D

Slide 115 / 149

EXAMPLE


9 6

x 5

Ans

wer

Slide 116 / 149

EXAMPLE


9 6

x 5


Ans

wer



3

x + 2x + 1

x - 1

Ans

wer

Slide 117 / 149


3

x + 2x + 1

x - 1


Ans

wer



x + 4

x - 2

5

4

Ans

wer

Slide 118 / 149


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 119 / 149

EXAMPLEFind RS.

R S

Q

T

16

x 8

Ans

wer

Slide 120 / 149

EXAMPLEFind RS.

R S

Q

T

16

x 8


Ans

wer

Since we are dealing with measurement, we only want the positive answer:



1

x

3

Ans

wer

Slide 121 / 149


1

x

3


Ans

wer



x12

24

Ans

wer

Slide 122 / 149


x12

24


Ans

wer


Equations of a Circle


Slide 123 / 149

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 124 / 149

EXAMPLE

Write the equation of the circle.

4 Ans

wer

Slide 125 / 149

page1svg

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 126 / 149

EXAMPLE

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

Ans

wer

Slide 127 / 149

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.

Ans

wer

Slide 128 / 149

EXAMPLE

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


Ans

wer

Then substitute the center and radius into the standard equation of a circle:

First, we need to find the length of the radius:


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 129 / 149

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

Ans

wer

Slide 130 / 149

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


Ans

wer 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

Ans

wer

Slide 131 / 149

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) Ans

wer

Slide 132 / 149

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?A

nsw

er

Slide 133 / 149

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

Ans

wer

Slide 134 / 149

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)

Ans

wer

Slide 135 / 149

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 136 / 149

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 137 / 149

page1svg

63 Which arc borders the minor sector?

A

B A

BC

D

Ans

wer

Slide 138 / 149

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

Ans

wer

Slide 139 / 149

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 140 / 149

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

450

AB

C

r= 3 Ans

wer

Slide 141 / 149

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

Ans

wer

Slide 142 / 149

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

Ans

wer

Slide 143 / 149

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

Ans

wer

Slide 144 / 149

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

Ans

wer

Slide 145 / 149

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

Ans

wer

Slide 146 / 149

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

Ans

wer

Slide 147 / 149

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?

Ans

wer

Slide 148 / 149

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?

Ans

wer

Slide 149 / 149

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


Geometry Circles - Currituck County Schools

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Transcript of Geometry Circles - Currituck County Schools