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1 This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: New Jersey Center for Teaching and Learning Progressive Mathematics Initiative

2 Geometry Circles

3 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

4 Parts of a Circle Return to the table of contents

5 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

6 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)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)A). (circle A or A.

7 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 &

8 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

9 1A diameter of a circle is the longest chord of the circle. True False [This object is a pull tab] Answer True

10 2A radius of a circle is a chord of a circle. True False [This object is a pull tab] Answer False

11 3Two radii of a circle always equal the length of a diameter of a circle. True False [This object is a pull tab] Answer True

12 4If 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

13 5How many diameters can be drawn in a circle? A 1 B 2 C 4 D infinitely many [This object is a pull tab] Answer D

14 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.

15 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

16 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)

17 6How many common tangent lines do the circles have? [This object is a pull tab] Answer 4

18 7How many common tangent lines do the circles have? [This object is a pull tab] Answer 1

19 8How many common tangent lines do the circles have? [This object is a pull tab] Answer 2

20 9How many common tangent lines do the circles have? [This object is a pull tab] Answer 0

21 Using the diagram below, match the notation with the term that best describes it: A C D E F G B. center radiuschord 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

22 Angles & Arcs Return to the table of contents

23 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.)

24 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

25 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

26 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 [This object is a pull tab] Answer

27 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. Name another semicircle. [This object is a pull tab] Answer

28 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 = 3200

29 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 C A T r arc length of = 3600 CT arc length of = or

30 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 arc length of = = cm

31 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.A. SY A.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 = = 9 1 = in

32 10In circle C where is a diameter, find 1350 A C B D 15 in [This object is a pull tab] Answer

33 11In circle C, where is a diameter, find 1350 A C B D 15 in [This object is a pull tab] Answer

34 12In circle C, where is a diameter, find 1350 A C B D 15 in [This object is a pull tab] Answer

35 13In circle C can it be assumed that AB is a diameter? Yes No 1350 A C B D [This object is a pull tab] Answer Yes

36 14Find the length of 450 A C 3 cm B [This object is a pull tab] Answer

37 15Find the circumference of circle T. T cm [This object is a pull tab] Answer

38 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

39 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 + =

40 EXAMPLE A result of a survey about the ages of people in a city are shown. Find the indicated measures. > S U V R T [This object is a pull tab] Answer = = = 1300 = = 2300 = = 2700

41 Match the type of arc and it's measure to the given arcs below: T S R Q minor arcmajor arc semicircle [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.

42 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 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.

43 17 True False A B C D [This object is a pull tab] Answer True

44 18 True False 850 M N L P [This object is a pull tab] Answer False

45 900 19Circle 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

46 20Two concentric circles always have congruent radii. True False [This object is a pull tab] Answer False

47 21If two circles have the same center, they are congruent. True False [This object is a pull tab] Answer False

48 22Tanny 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

49 Chords, Inscribed Angles & Polygons Return to the table of contents

50 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.

51 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.

52 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,

53 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"

54 If, then point Y and any line segment, or ray, that contains Y, bisects BISECTING ARCS C X Z Y

55 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 = = 720 = = x = 80 - x 10x = 80 x = 8

56 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

57 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 = 3xCU = 2(3) = 6 3 = x

58 23In circle R, and. Find A B C D R [This object is a pull tab] Answer 1080

59 24Given 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

60 25Given: 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

61 26AH is a diameter of the circle. True False A S H M T [This object is a pull tab] Answer False

62 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.

63 THEOREM: The measure of an inscribed angle is half the measure of its intercepted arc. C A T

64 EXAMPLE Q R T S P Findand [This object is a pull tab] Answer

65 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

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

67 27Given circle C below, find D E C A B [This object is a pull tab] Answer 500

68 28Given circle C below, find D E C A B [This object is a pull tab] Answer 1100

69 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

70 30Find X Y Z P. [This object is a pull tab] Answer 900

71 31In a circle, two parallel chords on opposite sides of the center have arcs which measure 1000 and Find the measure of one of the arcs included between the chords. [This object is a pull tab] Answer 700

72 32Given circle O, find the value of x.. O A B C D x 300 [This object is a pull tab] Answer 1200

73 33Given circle O, find the value of x.. O A B C D x [This object is a pull tab] Answer 1200

74 In the circle below, andFind, and Try This P S Q T [This object is a pull tab] Answer

75 INSCRIBED POLYGONS A polygon is inscribed if all its vertices lie on a circle.... inscribed triangle.... inscribed quadrilateral

76 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.

77 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

78 EXAMPLE Find the value of each variable: 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

79 34The value of x is A B C D C B A D x y [This object is a pull tab] Answer B

80 35In the diagram, is a central angle and. What is ? A B C D. B A D C [This object is a pull tab] Answer B

81 36What is the value of x? A 5 B 10 C 13 D 15 E F G (12x + 40)0 (8x + 10)0 [This object is a pull tab] Answer A

82 Tangents & Secants Return to the table of contents

83 **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. BB Line is tangent to circle X iff would be the point of tangency. l l

84 Verify A Line is Tangent to a Circle. T P S } Given:is a radius of circle P Is tangent to circle P? [This object is a pull tab] Answer Since = 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.

85 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 = AB r2 = (50 + r) r2 = r r = 100r = 100r 39 = r So, r = 39 ft.

86 THEOREM: Tangent segments from a common external point are congruent. R A T P. If AR and AT are tangent segments, then

87 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

88 37 AB is a radius of circle A. Is BC tangent to circle A? Yes No. B C A } [This object is a pull tab] Answer No

89 38S is a point of tangency. Find the radius r of circle T. A31.7 B60 C14 D3.5. T S R r r 48 cm 36 cm [This object is a pull tab] Answer C

90 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

91 40AB, 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

92 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.

93 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

94 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 tangentstwo secants W X Y Z 3

95 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

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

97 EXAMPLE Find the value of x x [This object is a pull tab] Answer x = 1/2 ( ) x = 1430

98 41Find the value of x. C H D F x E [This object is a pull tab] Answer

99 42Find the value of x. 340 (x + 6)0 (3x - 2)0 [This object is a pull tab] Answer

100 43Find A B 650 [This object is a pull tab] Answer

101 44Find [This object is a pull tab] Answer

102 45Find the value of x. x [This object is a pull tab] Answer

103 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.

104 46Find the value of x x0 [This object is a pull tab] Answer First find the minor arc.

105 47Find the value of x. x [This object is a pull tab] Answer First find the major arc.

106 48Find the value of x x0 500 [This object is a pull tab] Answer Find the major arc.

107 49Find the value of x (5x + 10)0 [This object is a pull tab] Answer Find the major arc.

108 50Find the value of x. (2x - 30)0 300 x [This object is a pull tab] Answer

109 Segments & Circles Return to the table of contents

110 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

111 EXAMPLE Find the value of x. 5 5 x 4 [This object is a pull tab] Answer

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

113 51Find the value of x x [This object is a pull tab] Answer

114 52Find 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

115 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

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

117 53Find the value of x. 3 x + 2 x + 1 x - 1 [This object is a pull tab] Answer

118 54Find the value of x. x + 4 x [This object is a pull tab] Answer

119 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

120 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:

121 55Find the value of x. 1 x 3 [This object is a pull tab] Answer

122 56Find the value of x. x [This object is a pull tab] Answer

123 Equations of a Circle Return to the table of contents

124 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.

125 EXAMPLE Write the equation of the circle. 4 [This object is a pull tab] Answer x2 + y2 = (4)2 x2 + y2 = 16

126 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.

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

128 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:

129 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 First plot the center and move 6 places in each direction. Then draw the circle.

130 57What 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 = [This object is a pull tab] Answer A

131 58What 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 [This object is a pull tab] Answer D

132 59What 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

133 60What is the radius of (x - 4)2 + (y - 2)2 = 64? [This object is a pull tab] Answer r=8

134 61The standard equation of a circle is (x - 2)2 + (y + 1)2 = 16. What is the diameter of the circle? A2 B4 C8 D16 [This object is a pull tab] Answer C

135 62Which 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

136 Return to the table of contents Area of a Sector

137 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

138 63Which arc borders the minor sector? A B A B C D [This object is a pull tab] Answer A

139 64Which arc borders the major sector? A B W X Y Z [This object is a pull tab] Answer B

140 Lets think about the formula... T he 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.

141 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

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

143 65Find 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

144 66Find 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

145 67What is the central angle for the major sector of the circle? C A G 15 cm 1200 [This object is a pull tab] Answer

146 68Find 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

147 69The sum of the major and minor sectors' areas is equal to the total area of the circle. True False [This object is a pull tab] Answer True

148 70A 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

149 71You 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


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