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Geometry Circleswww.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 Return to the 

table of 

contents

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 A A 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] Answer &

The relationship between the diameter and the radius A The measure of the diameter, d , is 
twice the measure of the radius, r . Therefore, or M C T If then what is the length of , In . A what is the length of [This object is a pull tab] Answer AC = 5 TC = 10

1 A diameter of a circle is the longest chord of the circle. TrueFalse [This object is a pull tab] Answer True

2 A radius of a circle is a chord of a circle. TrueFalse [This object is a pull tab] Answer False

3 Two radii of a circle always equal the length of a diameter of a circle. TrueFalse [This object is a pull tab] Answer True

4 If the radius of a circle measures 3.8 meters, what is the measure of the diameter? [This object is a pull tab] Answer 7.6 m

5 How many diameters can be drawn in a circle? A1 B 2 C 4 D infinitely many [This object is a pull tab] Answer 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.

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

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? [This object is a pull tab] Answer 4

7 How many common tangent lines do the circles have? [This object is a pull tab] Answer 1

8 How many common tangent lines do the circles have? [This object is a pull tab] Answer 2

9 How many common tangent lines do the circles have? [This object is a pull tab] Answer 0

Using the diagram below, match the notation with the term that 
best describes it: A C D E F G . . . . . . B . center radius chord diameter secant tangent point of tangency common tangent [This object is a pull tab] Answer Center Common Tangent Chord Secant Tangent Point of Tangency Radius Diameter

Angles & Arcs Return to the 

table of 

contents

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 degrees 2) 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 A T H S . . In , is the central 
angle. Name another central angle. [This object is a pull tab] Answer

M A T H S . . 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. MA Highlight [This object is a pull tab] Answer

M A T H S . . 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 [This object is a pull tab] Answer

M A T H S . . 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. Na me another semicircle. [This object is a pull tab] Answer

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 CT arc 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 [This object is a pull tab] Answer CT CT arc length of = 3600 . = 600 3600 . 8.38 cm

EXAMPLE S A Y 4.19 in 400 A In , 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 SY A [This object is a pull tab] Answer arc length of = 3600 SY SY = 3600 400 4.19 4.19 = 9 1 = 37.71 in

10 In circle C where is a diameter, find 1350 A C B D 15 in [This object is a pull tab] Answer

11 In circle C, where is a diameter, find 1350 A C B D 15 in [This object is a pull tab] Answer

12 In circle C, where is a diameter, find 1350 A C B D 15 in [This object is a pull tab] Answer

13 In circle C can it be assumed that AB is a diameter? YesNo 1350 A C B D [This object is a pull tab] Answer Yes

14 Find the length of 450 A C 3 cm B [This object is a pull tab] Answer

15 Find the circumference of circle T. T 750 6.82 cm [This object is a pull tab] Answer

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] Answer 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 + =

EXAMPLE A 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. [This object is a pull tab] Answer = 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 S R Q minor arc major arc semicircle 1200 2400 1800 1600 800 [This object is a pull tab] Teacher Notes 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 F 550 550 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 TrueFalse 1800 700 400 A B C D [This object is a pull tab] Answer True

18 TrueFalse 850 M N L P [This object is a pull tab] Answer False

900 19 Circle P has a radius of 3 and has a measure of . What is the length of ? A B C D P A B [This object is a pull tab] Answer A

20 Two concentric circles always have congruent radii. TrueFalse [This object is a pull tab] Answer False

21 If two circles have the same center, they are congruent. TrueFalse [This object is a pull tab] Answer False

22 Tanny cuts a pie into 6 congruent pieces. What is the measure of the central angle of each piece? [This object is a pull tab] Answer

Chords, Inscribed 

Angles & Polygons Return to the 

table of 

contents

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 S Q 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 B C D E . (9x)0 (80 - x)0 and , , Find: [This object is a pull tab] Answer = 9(8) = 720 = 80 - 8 = 720 = 720 + 720 = 1440 9x = 80 - x 10x = 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 D E A F B 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, [This object is a pull tab] Answer 2x = 5x - 9 9 = 3x  CU = 2(3) = 6 3 = x

23 In circle R , and . Find A B C D R . 1080 [This object is a pull tab] Answer 1080

24 Given circle C below, the length of is: A 5 B 10 C 15 D 20 D B F C . 10 A [This object is a pull tab] Answer 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 S Q T P W . V [This object is a pull tab] Answer C

26 AH is a diameter of the circle. True False A S H M 3 3 5 T [This object is a pull tab] Answer False

INSCRIBED ANGLES D O G 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 [This object is a pull tab] Answer

THEOREM: If two inscribed angles of a circle intercept the same arc, 
then the angles are congruent. D C B A since they both 
intercept

In a circle, parallel chords intercept congruent arcs. O B . A D C In circle O , if , then , then In circle O , if

27 Given circle C below, find D E C A B . 1000 350 [This object is a pull tab] Answer 500

28 Given circle C below, find D E C A B . 1000 350 [This object is a pull tab] Answer 1100

29 Given the figure below, which pairs of angles are 
congruent? A B C D R S U T [This object is a pull tab] Answer A

30 Find X Y Z P . [This object is a pull tab] Answer 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. [This object is a pull tab] Answer 700

32 Given circle O, find the value of x. . O A B C D x 300 [This object is a pull tab] Answer 1200

33 Given circle O, find the value of x. . O A B C D x 1000 350 [This object is a pull tab] Answer 1200

In the circle below, and Find , and Try This P S 1 2 3 4 Q T [This object is a pull tab] Answer

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

EXAMPLE Find the value of each variable: 2a 2a 4b 2b L K J M [This object is a pull tab] Answer 2a + 2a = 180 4a = 180 a = 450 4b + 2b = 180 6b = 180 b = 300

34 The value of x isA B C D 1500 980 1120 1800 C B A D x y 680 820 [This object is a pull tab] Answer B

35 In the diagram, is a central angle and . What is ? 150 300 600 1200 A B C D . B A D C 1200 600 300 150 [This object is a pull tab] Answer B

36 What is the value of x? A5 B 10 C 13 D 15 E F G (12x + 40)0 (8x + 10)0 [This object is a pull tab] Answer A

Tangents & Secants Return to the 

table of 

contents

**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 l Line is tangent to circle X iff would be the point of tangency. B B Line 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? [This object is a pull tab] Answer 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 . [This object is a pull tab] Answer AC2 + BC2 = AB2 802 + r2 = (50 + r)2 6400 + r2 = r2 + 100r + 2500 6400 = 100r + 2500 3900 = 100r 39 = r So, 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

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 [This object is a pull tab] Answer 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 } [This object is a pull tab] Answer 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 S R r r 48 cm 36 cm [This object is a pull tab] Answer 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 [This object is a pull tab] Answer

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 A C D O [This object is a pull tab] Answer

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 R 2 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 P Q M 2 . two tangents two secants W X Y Z 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 1 2

EXAMPLE Find the value of x. D C A B x0 760 1780 [This object is a pull tab] Answer

EXAMPLE Find the value of x. 1300 x0 1560 [This object is a pull tab] Answer x = 1/2 (1300 + 1560) x = 1430

41 Find the value of x. C H D F x0 780 420 E [This object is a pull tab] Answer

42 Find the value of x. 340 (x + 6)0 (3x - 2)0 [This object is a pull tab] Answer

43 Find A B 650 [This object is a pull tab] Answer

44 Find 1 2600 [This object is a pull tab] Answer

45 Find the value of x. x 122.50 450 [This object is a pull tab] Answer

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 [This object is a pull tab] Answer First find the minor arc.

46 Find the value of x. 2200 x0 [This object is a pull tab] Answer First find the minor arc.

47 Find the value of x. x0 1000 [This object is a pull tab] Answer First find the major arc.

48 Find the value of x x0 500 [This object is a pull tab] Answer Find the major arc.

49 Find the value of x. 1200 (5x + 10)0 [This object is a pull tab] Answer Find the major arc.

50 Find the value of x. (2x - 30)0 300 x [This object is a pull tab] Answer

Segments & Circles Return to the 
table of 
contents

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 [This object is a pull tab] Answer

EXAMPLE Find ML & JK. x + 2 x + 4 x x + 1 M K J L [This object is a pull tab] Answer ML = (2 + 2) +( 2 + 1) = 7 JK = 2 + (2 + 4) = 8

51 Find the value of x. 18 9 16 x [This object is a pull tab] Answer

52 Find the value of x. A-2 B 4 C 5 D 6 x 2 2x + 6 x [This object is a pull tab] Answer 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

EXAMPLE Find the value of x. 9 6 x 5 [This object is a pull tab] Answer

53 Find the value of x. 3 x + 2 x + 1 x - 1 [This object is a pull tab] Answer

54 Find the value of x. x + 4 x - 2 5 4 [This object is a pull tab] Answer

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 C D

EXAMPLE Find RS. R S Q T 16 x 8 [This object is a pull tab] Answer Since we are dealing with 
measurement, we only want 
the positive answer:

55 Find the value of x. 1 x 3 [This object is a pull tab] Answer

56 Find the value of x. x 12 24 [This object is a pull tab] Answer

Equations of a 

Circle Return to the 

table of 

contents

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 [This object is a pull tab] Answer 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.

EXAMPLE Write the standard equation of a circle with 
center (-2, 3) & radius 3.8. [This object is a pull tab] Answer

EXAMPLE The point (-5, 6) is on a circle with center (-1, 3). Write the 
standard equation of the circle. [This object is a pull tab] Answer 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.

57 What is the standard equation of the circle below? AB 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 = 400 10 [This object is a pull tab] Answer A

58 What is the standard equation of the circle? AB 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 [This object is a pull tab] Answer 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) [This object is a pull tab] Answer B

60 What is the radius of (x - 4)2 + (y - 2)2 = 64? [This object is a pull tab] Answer 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 [This object is a pull tab] Answer 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) [This object is a pull tab] Answer D

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? AB A B C D [This object is a pull tab] Answer A

64 Which arc borders the major sector? AB W X Y Z [This object is a pull tab] Answer B

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 Sector 1. Use the formula: when θ is in degrees 450 A B C r=3 [This object is a pull tab] Answer

Example: Find the Area of the major sector. C A T 8 cm 600 [This object is a pull tab] Answer cm2

65 Find the area of the minor sector of the circle. Round your answer to the nearest hundredth. C A T 5.5 cm 300 [This object is a pull tab] Answer 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 [This object is a pull tab] Answer cm2

67 What is the central angle for the major sector of the circle? C A G 15 cm 1200 [This object is a pull tab] Answer

68 Find the area of the major sector. Round to the nearest hundredth. C A G 15 cm 1200 [This object is a pull tab] Answer cm2

69 The sum of the major and minor sectors' areas is equal to the total area of the circle. TrueFalse [This object is a pull tab] Answer 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? [This object is a pull tab] Answer 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? [This object is a pull tab] Answer