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New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative

Geometry

Circles

www.njctl.org

2014-03-31

Table of Contents

Parts of a Circle

Angles & Arcs

Chords, Inscribed Angles & Polygons

Segments & Circles

Equations of a Circle

Click on a topic to go to that section

Tangents & Secants

Area of a Sector

Parts of a Circle

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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

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

.

A

M

C

R

T

is the diameter of circle A

is a chord of circle AA 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?

Answ

er

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

Answ

er

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

True

False

Answ

er

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

True

False

Answ

er

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

True

False

Answ

er

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

Answ

er

5 How many diameters can be drawn in a circle?

A 1

B 2

C 4

D infinitely many

Answ

er

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.

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 circles1 point

concentric circles

..

...

no points

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)

6 How many common tangent lines do the circles have?

Answ

er

7 How many common tangent lines do the circles have?

Answ

er

8 How many common tangent lines do the circles have?

Answ

er

9 How many common tangent lines do the circles have?

Answ

er

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

Answ

er

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.)

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.

Answ

er

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

Answ

er

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

Answ

er

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.

Answ

er

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

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

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

Answ

er

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

Answ

er

10 In circle C where is a diameter, find

1350

A

C

B

D

15 in

Answ

er

11 In circle C, where is a diameter, find

1350

A

C

B

D

15 in

Answ

er

12 In circle C, where is a diameter, find

1350

A

C

B

D

15 in

Answ

er

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

Yes

No 1350

A

C

B

D

Answ

er

14 Find the length of

450

A

C3 cm

B

Answ

er

15 Find the circumference of circle T.

T

750

6.82 cm

Answ

er

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

Answ

er

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

+=

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

300900

800600

1000

T

1.

2.

3.

4.

Answ

er

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

Not

es

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.

17

True

False1800

700400

A

B

C

D

Answ

er

18

True

False 850

M

N

L

P

Answ

er

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

Answ

er

20 Two concentric circles always have congruent radii.

True

False

Answ

er

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

True

False

Answ

er

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

Answ

er

Chords, Inscribed Angles & Polygons

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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.

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.

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,

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"

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

BISECTING ARCS

C

X

Z

Y

Find:

,, and

EXAMPLE

A

BC

D

E

. (9x)0

(80 - x)0

and, ,

Find:

Answ

er

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

EXAMPLE

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

.Q

R

S

T

U

V

2x

5x - 9

C

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

Answ

er

23 In circle R, and . FindA

B

C

D

R.1080

Answ

er 1080

24 Given circle C below, the length of is:

A 5

B 10

C 15

D 20

D B

F

C.10

A

Answ

er D

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

A 1

B 7

C 20

D 8

R

SQ

T

P

W

.V

Answ

er C

26 AH is a diameter of the circle.

True

False

A

S

H

M

3

3

5

T

Answ

er

False

INSCRIBED ANGLES

D

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.

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

C

A

T

EXAMPLE

Q R

T S

P. 500

480

Find and

Answ

er

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

D

C

B

Asince they both intercept

In a circle, parallel chords intercept congruent arcs.

O

B

.A

DC

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

27 Given circle C below, find

D E

C

A B

. 1000

350 Answ

er

28 Given circle C below, find

D E

C

A B

. 1000

350

Answ

er 1100

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

A

B

C

D

RS

U

T

Answ

er

30 Find

X

Y

Z

P.

Answ

er 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.

Answ

er

32 Given circle O, find the value of x.

.O

A B

C D

x

300

Answ

er 1200

33 Given circle O, find the value of x.

.O

A B

C D

x

1000

350

Answ

er 1200

In the circle below, and Find

, and

Try This

P

S

1

2

3

4

Q

T

Answ

er

INSCRIBED POLYGONS

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

.

.

.

inscribed triangle

.

.

.

.

inscribed quadrilateral

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.

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

EXAMPLE

Find the value of each variable:

2a

2a4b

2bL

K

J

M

Answ

er

34 The value of x is

A

B

C

D

1500

980

1120

1800

C

B

A

Dx

y

680

820

Answ

er

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

Answ

er

36 What is the value of x?

A 5

B 10

C 13

D 15

E

F G

(12x + 40)0

(8x + 10)0

Answ

er

Tangents & Secants

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**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

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?

Answ

er

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.

Answ

er

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

R

A

T

P.

If AR and AT are tangent segments, then

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 Answ

er

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

Yes

No

.B C

A

60

25

67

}

Answ

er

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

Answ

er

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

A

D

B

C.25

3x - 8

Answ

er

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 Answ

er

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.

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

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

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

EXAMPLE

Find the value of x.

D

C

A

B

x0 760 1780

Answ

er

EXAMPLE

Find the value of x.

1300

x0

1560

Answ

er

41 Find the value of x.

C

H

DFx0

780

420E

Answ

er

42 Find the value of x.

340(x + 6)0

(3x - 2)0

Answ

er

43 Find

A

B

650

Answ

er

44 Find

12600

Answ

er

45 Find the value of x.

x122.50

450

Answ

er

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

Answ

er

46 Find the value of x.

2200

x0

Answ

er

47 Find the value of x.

x01000 An

swer

48 Find the value of x

x0500

Answ

er

49 Find the value of x.

1200

(5x + 10)0

Answ

er

50 Find the value of x.

(2x - 30)0

300 x

Answ

er

Segments & Circles

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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

EXAMPLE

Find the value of x.

5

5

x

4

Answ

er

EXAMPLE

Find ML & JK.

x + 2

x + 4

x x + 1

M K

J

L

Answ

er

51 Find the value of x.

189

16

x

Answ

er

52 Find the value of x.

A -2

B 4

C 5

D 6x

2

2x + 6

x

Answ

er

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

EXAMPLE

Find the value of x.

9 6

x 5

Answ

er

53 Find the value of x.

3

x + 2x + 1

x - 1

Answ

er

54 Find the value of x.

x + 4

x - 2

5

4

Answ

er

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

EXAMPLE

Find RS.

R S

Q

T

16

x 8

Answ

er

55 Find the value of x.

1

x

3

Answ

er

56 Find the value of x.

x12

24

Answ

er

Equations of a Circle

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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.

EXAMPLE

Write the equation of the circle.

4 Answ

er

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.

EXAMPLE

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

Answ

er

EXAMPLE

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

Answ

er

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.

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

Answ

er

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

Answ

er

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) Answ

er

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

Answ

er

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

Answ

er

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)

Answ

er

Return to the table of contents

Area of a Sector

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

63 Which arc borders the minor sector?

A

B A

BC

D

Answ

er

64 Which arc borders the major sector?

A

B

W

X

YZ

Answ

er

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

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

450

AB

C

r=3

Answ

er

Example:

Find the Area of the major sector.

C

A

T

8 cm

600 Answ

er

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

C

A

T5.5 cm 300

Answ

er

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

C

A

T

12 cm

850

Answ

er

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

C

A

G

15 cm

1200

Answ

er

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

C

A

G

15 cm

1200

Answ

er

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

True

False

Answ

er

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?

Answ

er

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?

Answ

er