Slide 1 / 150 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org 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: www.njctl.org
Slide 2 / 150 Geometry Circles 2014-06-03 www.njctl.org
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
Slide 4 / 150 Parts of a Circle Return to the table of contents
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
. 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.
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?
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
Slide 9 / 150 1 A diameter of a circle is the longest chord of the circle. True False
Slide 10 / 150 2 A radius of a circle is a chord of a circle. True False
Slide 11 / 150 3 Two radii of a circle always equal the length of a diameter of a circle. True False
Slide 12 / 150 4 If the radius of a circle measures 3.8 meters, what is the measure of the diameter?
Slide 13 / 150 5 How many diameters can be drawn in a circle? A 1 B 2 C 4 D infinitely many
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.
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
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)
Slide 17 / 150 6 How many common tangent lines do the circles have?
Slide 18 / 150 7 How many common tangent lines do the circles have?
Slide 19 / 150 8 How many common tangent lines do the circles have?
Slide 20 / 150 9 How many common tangent lines do the circles have?
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
Slide 22 / 150 Angles & Arcs Return to the table of contents
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.)
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.
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.
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
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.
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 360 0 minus the measure of the central angle. B 40 0 D. A 40 0 G 36 0-40 0 = 320 0
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 360 0. C arc length of CT = CT 36 0 A r or T arc length of CT =. CT 36 0
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
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
Slide 32 / 150 10 In circle C where is a diameter, find B D 135 0 C 15 in A
Slide 33 / 150 11 In circle C, where is a diameter, find B D A 135 0 C 15 in
Slide 34 / 150 12 In circle C, where is a diameter, find B D A 135 0 C 15 in
Slide 35 / 150 13 In circle C can it be assumed that AB is a diameter? Yes B No D 135 0 C A
Slide 36 / 150 14 Find the length of A B C 45 0 3 cm
Slide 37 / 150 15 Find the circumference of circle T. 75 0 T 6.82 cm
Slide 38 / 150 16 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
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.
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 >65 2. 30 0 90 0 3. 17-44 10 0 60 0 80 0 U 4. R 45-64 15-17 V
Slide 41 / 150 Match the type of arc and it's measure to the given arcs below: T Q S 12 0 80 0 60 0 R Teacher Notes m inor arc m ajor arc sem icircle 80 0 12 0 16 0 18 0 24 0
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 55 0 55 0 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.
Slide 43 / 150 17 A True False 18 0 70 0 40 0 B C D
Slide 44 / 150 18 True False 85 0 L M P N
Slide 45 / 150 19 Circle P has a radius of 3 and has a measure of 90 0. What is the length of? A A B C P D B
Slide 46 / 150 20 Two concentric circles always have congruent radii. True False
Slide 47 / 150 21 If two circles have the same center, they are congruent. True False
Slide 48 / 150 22 Tanny cuts a pie into 6 congruent pieces. What is the measure of the central angle of each piece?
Slide 49 / 150 Chords, Inscribed Angles & Polygons Return to the table of contents
Slide 50 / 150 Click on the link below and complete the labs before the Chords lesson. Lab - Properties of Chords
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.
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
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
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"
Slide 55 / 150 BISECTING ARCS X C Y If, then point Y and any line segment, or ray, that contains Y, bisects Z
B C EXAMPLE Slide 56 / 150 Find:,, and. (9x) 0 A D E (80 - x) 0
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
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,
Slide 59 / 150 23 In circle R, and. Find A 108 0 B R. C D
Slide 60 / 150 24 Given circle C below, the length of is: A 5 B 10 D 10 A B C 15 D 20. C F
Slide 61 / 150 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 Q V. P W R S T
Slide 62 / 150 26 AH is a diameter of the circle. A True 3 5 False M 3 S T H
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
Slide 64 / 150 THEOREM: The measure of an inscribed angle is half the measure of its intercepted arc. C A T
Slide 65 / 150 EXAMPLE Find and Q. 500 P R 48 0 T S
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
Slide 67 / 150 In a circle, parallel chords intercept congruent arcs. C O. D In circle O, if, then A B
Slide 68 / 150 27 Given circle C below, find D A E. C 10 0 35 0 B
Slide 69 / 150 28 Given circle C below, find D A E. C 10 0 35 0 B
Slide 70 / 150 29 Given the figure below, which pairs of angles are congruent? S A C U R B D T
Slide 71 / 150 30 Find X Y. P Z
Slide 72 / 150 31 In a circle, two parallel chords on opposite sides of the center have arcs which measure 100 0 and 120 0. Find the measure of one of the arcs included between the chords.
Slide 73 / 150 32 Given circle O, find the value of x. x 30 0 C A Ȯ B D
Slide 74 / 150 33 Given circle O, find the value of x. 10 0 A B 35 0. O C D x
Try This Slide 75 / 150 In the circle below, and Find, and P 2 1 3 Q T 4 S
Slide 76 / 150 INSCRIBED POLYGONS A polygon is inscribed if all its vertices lie on a circle.... inscribed triangle.. inscribed quadrilateral..
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
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
EXAMPLE Slide 79 / 150 Find the value of each variable: L 2a 2b M K 4b 2a J
Slide 80 / 150 34 The value of x is C A 150 0 68 0 B C 98 0 112 0 B x 82 0 D D 180 0 A y
Slide 81 / 150 35 In the diagram, is a central angle and. What is? A 15 0 A B C D 30 0 60 0 120 0 B D. C
Slide 82 / 150 36 What is the value of x? E A 5 B 10 0 (12x + 40) C 13 D 15 F 0 (8x + 10) G
Slide 83 / 150 Tangents & Secants Return to the table of contents
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
S } T 35 37 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?
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
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
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
Slide 89 / 150 37 AB is a radius of circle A. Is BC tangent to circle A? Yes No 25 A B. 60 }67 C
Slide 90 / 150 38 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
Slide 91 / 150 39 In circle C, DA is tangent at A and DB is tangent at B. Find x. A C. 25 D B 3x - 8
Slide 92 / 150 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. O F C D A
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.
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
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
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
Slide 97 / 150 EXAMPLE Find the value of x. D C x 0 76 0 178 0 B A
EXAMPLE Slide 98 / 150 Find the value of x. 13 0 x 0 156 0
Slide 99 / 150 41 Find the value of x. C 78 0 E H x 0 D 42 0 F
Slide 100 / 150 42 Find the value of x. 0 (3x - 2) (x + 6) 0 34 0
Slide 101 / 150 43 Find B A 65 0
Slide 102 / 150 44 Find 1 26 0
Slide 103 / 150 45 Find the value of x. x 122.5 0 45 0
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 247 0 x 0 A
Slide 105 / 150 46 Find the value of x. Students type their answers here 22 0 x 0
Slide 106 / 150 47 Find the value of x. Students type their answers here x 0 10 0
Slide 107 / 150 48 Find the value of x Students type their answers here 50 0 x 0
Slide 108 / 150 49 Find the value of x. Students type their answers here (5x + 10) 0 12 0
Slide 109 / 150 50 Find the value of x. 0 (2x - 30) 30 0 x
Slide 110 / 150 Segments & Circles Return to the table of contents
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
Slide 112 / 150 EXAMPLE Find the value of x. 4 5 5 x
Find ML & JK. Slide 113 / 150 EXAMPLE M K J x x + 2 L x + 4 x + 1
Slide 114 / 150 51 Find the value of x. x 9 16 18
Slide 115 / 150 52 Find the value of x. A -2 B 4 C 5 D 6 2 x x 2x + 6
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
Slide 117 / 150 EXAMPLE Find the value of x. 9 6 x 5
Slide 118 / 150 53 Find the value of x. 3 x + 1 x + 2 x - 1
Slide 119 / 150 54 Find the value of x. 5 4 x - 2 x + 4
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
Slide 121 / 150 Find RS. EXAMPLE Q 16 R x S 8 T
Slide 122 / 150 55 Find the value of x. 3 1 x
Slide 123 / 150 56 Find the value of x. 24 12 x
Slide 124 / 150 Equations of a Circle Return to the table of contents
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.
Slide 126 / 150 EXAMPLE Write the equation of the circle. 4
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.
Slide 128 / 150 EXAMPLE Write the standard equation of a circle with center (-2, 3) & radius 3.8.
Slide 129 / 150 EXAMPLE The point (-5, 6) is on a circle with center (-1, 3). Write the standard equation of the circle.
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..
Slide 131 / 150 57 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 = 400 10
Slide 132 / 150 58 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
Slide 133 / 150 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)
Slide 134 / 150 60 What is the radius of (x - 4) 2 + (y - 2) 2 = 64?
Slide 135 / 150 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
Slide 136 / 150 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)
Slide 137 / 150 Area of a Sector Return to the table of contents
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
Slide 139 / 150 63 Which arc borders the minor sector? A B A C B D
Slide 140 / 150 64 Which arc borders the major sector? A B W X Z Y
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.
Slide 142 / 150 Finding the Area of a Sector 1. Use the formula: when θ is in degrees A r= 3 45 0 B C
Slide 143 / 150 Example: Find the Area of the major sector. C 8 cm A 60 0 T
Slide 144 / 150 65 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
Slide 145 / 150 66 Find the Area of the major sector for the circle. Round your answer to the nearest thousandth. C 12 cm A 85 0 T
Slide 146 / 150 67 What is the central angle for the major sector of the circle? 15 cm C A 12 0 G
Slide 147 / 150 68 Find the area of the major sector. Round to the nearest hundredth. 15 cm C A 12 0 G
Slide 148 / 150 69 The sum of the major and minor sectors' areas is equal to the total area of the circle. True False
Slide 149 / 150 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?
Slide 150 / 150 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?