# New Jersey Center for Teaching and Learning. Progressive Mathematics Initiative

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1 Slide 1 / 150 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative 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:

2 Slide 2 / 150 Geometry Circles

3 Slide 3 / 150 Table of Contents Click on a topic to go to that section Parts of a Circle Angles & Arcs Chords, Inscribed Angles & Polygons Tangents & Secants Segments & Circles Equations of a Circle Area of a Sector

5 Slide 5 / 150 A circle is the set of all points in a plane that are a fixed distance from a given point in the plane called the center. center

6 . Slide 6 / 150 The symbol for a circle is and is named by a capital letter placed by the center of the circle.. (circle A or. A) is a radius of. A. A B 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.

7 Slide 7 / 150 M R is a chord of circlea A chord is a segment that has its endpoints on the circle. A T C is the diameter of circle A 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?

8 Slide 8 / 150 The relationship between the diameter and radius the T The measure of the diameter, d, is twice the measure of the radius, r. A M Therefore, or C In. A If, then what is the length of what is the length of

9 Slide 9 / A diameter of a circle is the longest chord of the circle. True False

10 Slide 10 / A radius of a circle is a chord of a circle. True False

11 Slide 11 / Two radii of a circle always equal the length of a diameter of a circle. True False

12 Slide 12 / If the radius of a circle measures 3.8 meters, what is the measure of the diameter?

13 Slide 13 / How many diameters can be drawn in a circle? A 1 B 2 C 4 D infinitely many

14 A Slide 14 / 150 A secant of a circle is a line that intersects the circle at two points. line l is a secant of this circle. D B l A tangent is a line in the plane of a circle that intersects the circle at exactly one point (the point of tangency). E k 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 Slide 15 / 150 COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points. Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric... 2 points.. tangent circles 1 point no points. concentric circles

16 Slide 16 / 150 A Common Tangent is a line, ray, or segment that is tangent to 2 coplanar circles. Internally tangent (tangent line passes between them) Externally tangent (tangent line does not pass between them)

17 Slide 17 / How many common tangent lines do the circles have?

18 Slide 18 / How many common tangent lines do the circles have?

19 Slide 19 / How many common tangent lines do the circles have?

20 Slide 20 / How many common tangent lines do the circles have?

21 Slide 21 / 150 Using the diagram below, match the notation with the term that best describes it:. A. C.. B D. F.. G E center chord secant radius diameter tangent common tangent point of tangency

23 Slide 23 / 150 An ARC is an unbroken piece of a circle with endpoints on the circle.. A Arc of the circle or AB. B 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 360 o.)

24 Slide 24 / 150 S.. H A central angle is an angle whose vertex is the center of the circle. M T A In, is the central angle. Name another central angle.

25 Slide 25 / 150 If is less than 180 0, then the points on that lie in the interior of form the minor arc with endpoints M and H. S.. H M minor arc MA Highlight MA T A Name another minor arc.

26 Slide 26 / 150 major arc S.. H T M A Points M and A and all points of exterior to form a major arc, MSA Major arcs are the "long way" around the circle. Major arcs are greater than 180 o. Highlight Major arcs are named by their endpoints and a point on the arc. Name another major arc. MSA

27 Slide 27 / 150 S.. H M minor arc T A 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.

28 Slide 28 / 150 Measurement By A Central Angle The measure of a minor arc is the measure of its central angle. The measure of the major arc is minus the measure of the central angle. B 40 0 D. A 40 0 G = 320 0

29 Slide 29 / 150 The Length of the Arc Itself (AKA - Arc Length) Arc length is a portion of the circumference of a circle. Arc Length Corollary - In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to C arc length of CT = CT 36 0 A r or T arc length of CT =. CT 36 0

30 Slide 30 / 150 EXAMPLE In A, the central angle is 60 0 and the radius is 8 cm. Find the length of CT 8 cm A C 60 0 T

31 Slide 31 / 150 EXAMPLE In A, the central angle is 40 0 and the length of is 4.19 in. Find the circumference of A. SY S 4.19 in A 40 0 Y

32 Slide 32 / In circle C where is a diameter, find B D C 15 in A

33 Slide 33 / In circle C, where is a diameter, find B D A C 15 in

34 Slide 34 / In circle C, where is a diameter, find B D A C 15 in

35 Slide 35 / In circle C can it be assumed that AB is a diameter? Yes B No D C A

36 Slide 36 / Find the length of A B C cm

37 Slide 37 / Find the circumference of circle T T 6.82 cm

38 Slide 38 / In circle T, WY & XZ are diameters. WY = XZ = 6. If XY = 14 0, what is the length of YZ? A B C W T X D Z Y

39 Slide 39 / 150 ADJACENT ARCS 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. C.. A = + T.

40 Slide 40 / 150 EXAMPLE A result of a survey about the ages of people in a city are shown. Find the indicated measures. T 1. S > U 4. R V

41 Slide 41 / 150 Match the type of arc and it's measure to the given arcs below: T Q S R Teacher Notes m inor arc m ajor arc sem icircle

42 Slide 42 / 150 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. T D E R C F S 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 Slide 43 / A True False B C D

44 Slide 44 / True False 85 0 L M P N

45 Slide 45 / Circle P has a radius of 3 and has a measure of What is the length of? A A B C P D B

46 Slide 46 / Two concentric circles always have congruent radii. True False

47 Slide 47 / If two circles have the same center, they are congruent. True False

48 Slide 48 / Tanny cuts a pie into 6 congruent pieces. What is the measure of the central angle of each piece?

50 Slide 50 / 150 Click on the link below and complete the labs before the Chords lesson. Lab - Properties of Chords

51 Slide 51 / 150 When a minor arc and a chord have the same endpoints, we call the arc The Arc of the Chord. P C. Q is the arc of **Recall the definition of a chord - a segment with endpoints on the circle.

52 Slide 52 / 150 THEOREM: In a circle, if one chord is a perpendicular bisector of another chord, then the first chord is a diameter. T is the perpendicular bisector of. Therefore, is a diameter of the circle. S Q E Likewise, the perpendicular bisector of a chord of a circle passes through the center of a circle. P

53 Slide 53 / 150 THEOREM: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. C A is a diameter of the circle and is perpendicular to chord.x Therefore, S E

54 Slide 54 / 150 THEOREM: In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. B C iff A D *iff stands for "if and only if"

55 Slide 55 / 150 BISECTING ARCS X C Y If, then point Y and any line segment, or ray, that contains Y, bisects Z

56 B C EXAMPLE Slide 56 / 150 Find:,, and. (9x) 0 A D E (80 - x) 0

57 Slide 57 / 150 THEOREM: In the same circle, or congruent circles, two chords are congruent if and only if they are equidistant from the center. C A E. G D iff F B

58 Slide 58 / 150 EXAMPLE Given circle C, QR = ST = 16. Find CU. Q S U. 2x C 5x - 9 V R T Since the chords QR & ST are congruent, they are equidistant from C. Therefore,

59 Slide 59 / In circle R, and. Find A B R. C D

60 Slide 60 / Given circle C below, the length of is: A 5 B 10 D 10 A B C 15 D 20. C F

61 Slide 61 / 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 Q V. P W R S T

62 Slide 62 / AH is a diameter of the circle. A True 3 5 False M 3 S T H

63 Slide 63 / 150 INSCRIBED ANGLES Inscribed angles are angles whose vertices are in on the circle and whose sides are chords of the circle. O D G 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. Click on the link below and complete the lab. Lab - Inscribed Angles

64 Slide 64 / 150 THEOREM: The measure of an inscribed angle is half the measure of its intercepted arc. C A T

65 Slide 65 / 150 EXAMPLE Find and Q. 500 P R 48 0 T S

66 Slide 66 / 150 THEOREM: If two inscribed angles of a circle intercept the same arc, then the angles are congruent. D A B since they both intercept C

67 Slide 67 / 150 In a circle, parallel chords intercept congruent arcs. C O. D In circle O, if, then A B

68 Slide 68 / Given circle C below, find D A E. C B

69 Slide 69 / Given circle C below, find D A E. C B

70 Slide 70 / Given the figure below, which pairs of angles are congruent? S A C U R B D T

71 Slide 71 / Find X Y. P Z

72 Slide 72 / In a circle, two parallel chords on opposite sides of the center have arcs which measure and Find the measure of one of the arcs included between the chords.

73 Slide 73 / Given circle O, find the value of x. x 30 0 C A Ȯ B D

74 Slide 74 / Given circle O, find the value of x A B O C D x

75 Try This Slide 75 / 150 In the circle below, and Find, and P Q T 4 S

76 Slide 76 / 150 INSCRIBED POLYGONS A polygon is inscribed if all its vertices lie on a circle.... inscribed triangle.. inscribed quadrilateral..

77 Slide 77 / 150 THEOREM: If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. A L x. iff AC is a diameter of the circle. G

78 Slide 78 / 150 THEOREM: A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. E N C. N, E, A, and R lie on circle C iff A R

79 EXAMPLE Slide 79 / 150 Find the value of each variable: L 2a 2b M K 4b 2a J

80 Slide 80 / The value of x is C A B C B x 82 0 D D A y

81 Slide 81 / In the diagram, is a central angle and. What is? A 15 0 A B C D B D. C

82 Slide 82 / What is the value of x? E A 5 B 10 0 (12x + 40) C 13 D 15 F 0 (8x + 10) G

84 Slide 84 / 150 **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). l Line is tangent to circle X iff B would be the point of tangency. Click on the link below and complete the lab. l X.. B l Lab - Tangent Lines

85 S } T Slide 85 / 150 Verify A Line is Tangent to a Circle 12. P Given: is a radius of circle P Is tangent to circle P?

86 Slide 86 / 150 Finding the Radius of a Circle If B is a point of tangency, find the radius of circle C. A 80 ft 50 ft r r. C B

87 Slide 87 / 150 THEOREM: Tangent segments from a common external point are congruent. R P. A T If AR and AT are tangent segments, then

88 Slide 88 / 150 EXAMPLE Given: RS is tangent to circle C at S and RT is tangent to circle C at T. Find x. S 28 C. T 3x + 4 R

89 Slide 89 / AB is a radius of circle A. Is BC tangent to circle A? Yes No 25 A B. 60 }67 C

90 Slide 90 / S is a point of tangency. Find the radius r of circle T. A 31.7 B 60 C 14 T. r r 36 cm D 3.5 S 48 cm R

91 Slide 91 / In circle C, DA is tangent at A and DB is tangent at B. Find x. A C. 25 D B 3x - 8

92 Slide 92 / AB, BC, and CA are tangents to circle O. AD = 5, AC= 8, and BE = 4. Find the perimeter of triangle ABC. B E. O F C D A

93 Slide 93 / 150 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.

94 Slide 94 / 150 A Tangent and a Chord THEOREM: If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. A.. M. 2 1 R

95 Slide 95 / 150 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 tangent and a secant A 1 C B 2 two tangents P M. Q two secants X W 3 Y Z

96 Slide 96 / 150 THEOREM: If two chords intersect inside a circle, then the measure of each angle is half the sum of the intercepted arcs by the angle and vertical angle. M A 2 1 H T

97 Slide 97 / 150 EXAMPLE Find the value of x. D C x B A

98 EXAMPLE Slide 98 / 150 Find the value of x x

99 Slide 99 / Find the value of x. C 78 0 E H x 0 D 42 0 F

100 Slide 100 / Find the value of x. 0 (3x - 2) (x + 6)

101 Slide 101 / Find B A 65 0

102 Slide 102 / Find

103 Slide 103 / Find the value of x. x

104 Slide 104 / 150 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 x 0. B x 0 A

105 Slide 105 / Find the value of x. Students type their answers here 22 0 x 0

106 Slide 106 / Find the value of x. Students type their answers here x

107 Slide 107 / Find the value of x Students type their answers here 50 0 x 0

108 Slide 108 / Find the value of x. Students type their answers here (5x + 10)

109 Slide 109 / Find the value of x. 0 (2x - 30) 30 0 x

111 Slide 111 / 150 THEOREM: If two chords intersect inside a circle, then the products of the measures of the segments of the chords are equal. A C E D B

112 Slide 112 / 150 EXAMPLE Find the value of x x

113 Find ML & JK. Slide 113 / 150 EXAMPLE M K J x x + 2 L x + 4 x + 1

114 Slide 114 / Find the value of x. x

115 Slide 115 / Find the value of x. A -2 B 4 C 5 D 6 2 x x 2x + 6

116 Slide 116 / 150 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. B A E C D

117 Slide 117 / 150 EXAMPLE Find the value of x. 9 6 x 5

118 Slide 118 / Find the value of x. 3 x + 1 x + 2 x - 1

119 Slide 119 / Find the value of x. 5 4 x - 2 x + 4

120 Slide 120 / 150 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

121 Slide 121 / 150 Find RS. EXAMPLE Q 16 R x S 8 T

122 Slide 122 / Find the value of x. 3 1 x

123 Slide 123 / Find the value of x x

125 Slide 125 / 150 r x (x, y) y Let (x, y) be any point on a circle with center at the origin and radius, r. By the Pythagorean Theorem, x 2 + y 2 = r 2 This is the equation of a circle with center at the origin.

126 Slide 126 / 150 EXAMPLE Write the equation of the circle. 4

127 Slide 127 / 150 For circles whose center is not at the origin, we use the distance formula to derive the equation. (x, y). r (h, k) This is the standard equation of a circle.

128 Slide 128 / 150 EXAMPLE Write the standard equation of a circle with center (-2, 3) & radius 3.8.

129 Slide 129 / 150 EXAMPLE The point (-5, 6) is on a circle with center (-1, 3). Write the standard equation of the circle.

130 Slide 130 / 150 EXAMPLE The equation of a circle is (x - 4) 2 + (y + 2) 2 = 36. Graph the circle. We know the center of the circle is (4, -2) and the radius is 6.. First plot the center and move 6 places in each direction... Then draw the circle..

131 Slide 131 / What is the standard equation of the circle below? A B C D x 2 + y 2 = 400 (x - 10) 2 + (y - 10) 2 = 400 (x + 10) 2 + (y - 10) 2 = 400 (x - 10) 2 + (y + 10) 2 =

132 Slide 132 / What is the standard equation of the circle? A B C D (x - 4) 2 + (y - 3) 2 = 81 (x - 4) 2 + (y - 3) 2 = 9 (x + 4) 2 + (y + 3) 2 = 81 (x + 4) 2 + (y + 3) 2 = 9

133 Slide 133 / What is the center of (x - 4) 2 + (y - 2) 2 = 64? A (0,0) B (4,2) C (-4, -2) D (4, -2)

134 Slide 134 / What is the radius of (x - 4) 2 + (y - 2) 2 = 64?

135 Slide 135 / 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

136 Slide 136 / 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)

138 Slide 138 / 150 A sector of a circle is the portion of the circle enclosed by two radii and the arc that connects them. B Minor Sector Major Sector A C

139 Slide 139 / Which arc borders the minor sector? A B A C B D

140 Slide 140 / Which arc borders the major sector? A B W X Z Y

141 Slide 141 / 150 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.

142 Slide 142 / 150 Finding the Area of a Sector 1. Use the formula: when θ is in degrees A r= B C

143 Slide 143 / 150 Example: Find the Area of the major sector. C 8 cm A 60 0 T

144 Slide 144 / Find the area of the minor sector of the circle. Round your answer to the nearest hundredth. A C 5.5 cm 30 0 T

145 Slide 145 / Find the Area of the major sector for the circle. Round your answer to the nearest thousandth. C 12 cm A 85 0 T

146 Slide 146 / What is the central angle for the major sector of the circle? 15 cm C A 12 0 G

147 Slide 147 / Find the area of the major sector. Round to the nearest hundredth. 15 cm C A 12 0 G

148 Slide 148 / The sum of the major and minor sectors' areas is equal to the total area of the circle. True False

149 Slide 149 / 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?

150 Slide 150 / 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?

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