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Slide 1 / 150Slide 2 / 150New Jersey Center for Teaching and LearningGeometryProgressive Mathematics Initiative This material is made freely available at www.njctl.organd is intended for the non-commercial use ofstudents and teachers. These materials may not beused for any commercial purpose without the writtenpermission of the owners. NJCTL maintains itswebsite for the convenience of teachers who wish tomake their work available to other teachers,participate in a virtual professional learningcommunity, and/or provide access to coursematerials to parents, students and others.Circles2014-06-03www.njctl.orgClick to go to website:www.njctl.orgSlide 3 / 150Slide 4 / 150Table of ContentsClick on a topic to goto that sectionParts of a CircleParts of a CircleAngles & ArcsChords, Inscribed Angles & PolygonsTangents & SecantsSegments & CirclesReturn to thetable ofcontentsEquations of a CircleArea of a SectorSlide 5 / 150A circle is the set of all points in aplane that are a fixed distance froma given point in the plane calledthe center.centerSlide 6 / 150The symbol for a circle is . and is named by a capital letterplaced by the center of the circle.(circle A or. A)is a radius ofAB.AA radius (plural, radii) is a linesegment drawn from the centerof the circle to any point on thecircle. It follows from thedefinition of a circle that all radiiof a circle are congruent.

Slide 7 / 150is a chord of circle AACTThe relationship between the diameter andradiustheA chord is a segment that has itsendpoints on the circle.is the diameter of circleTAA diameter is a chord that goesthrough the center of the circle.All diameters of a circle arecongruent.The measure of the diameter, d, istwice the measure of the radius, r.AMCWhat are the radii in this diagram?Therefore,orIn . AIf, then what is the length ofAnswerRAnswerMSlide 8 / 150what is the length ofSlide 9 / 150A diameter of a circle is the longest chord of thecircle.2A radius of a circle is a chord of a circle.TrueFalseFalseAnswerTrueAnswerSlide 11 / 150Two radii of a circle always equal the length of adiameter of a circle.4If the radius of a circle measures 3.8 meters, whatis the measure of the diameter?TrueFalseAnswer3Slide 12 / 150Answer1Slide 10 / 150

Slide 13 / 150A1B2C4Dinfinitely manyA secant of a circle is a line thatintersects the circle at two points.AHow many diameters can be drawn in a circle?line l is a secant of this circle.BDAnswer5Slide 14 / 150ElA tangent is a line in the plane ofa circle that intersects the circleat exactly one point (the point oftangency).line k is a tangentkD is the point of tangency.tangent ray,, and the tangent segment,,are also called tangents. They must be part of atangent line.Note: This is not a tangent ray.Slide 15 / 150Slide 16 / 150COPLANAR CIRCLES are two circles in the same plane whichintersect at 2 points, 1 point, or no points.A Common Tangent is a line, ray, or segment that is tangent to 2coplanar circles.Coplanar circles that intersects in 1 point are called tangentcircles. Coplanar circles that have a common center are calledconcentric.2 points.tangentcircles1 pointconcentriccirclesInternally tangent(tangent linepassesbetween them)no pointsSlide 17 / 150Slide 18 / 1507How many common tangent lines do the circleshave?Answer6Externally tangent(tangent line doesnot pass betweenthem)How many common tangent lines do the circleshave?Answer.

Slide 19 / 150How many common tangent lines do the circleshave?Answer9How many common tangent lines do the circleshave?Answer8Slide 20 / 150Slide 21 / 150Slide 22 / 150Using the diagram below, match the notation with the term thatbest describes it:.B.A.EAngles & Arcs.F.GReturn to thetable on tangentpoint of tangencySlide 23 / 150Slide 24 / 150An ARC is an unbroken piece of a circle with endpointson the circle.AArc of the circle or AB.BArcs 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 thecenter of the circle.M. .HSTAIn,is the centralangle.Name another central angle.Answercenterchord

Slide 25 / 150Slide 26 / 150Ifis less than 1800, then the points onthat lie in the interior ofform the minor arc withendpoints M and H. .HAnswerHHighlight MAAnswerminor arc MA. .ATPoints M and A and all points ofexterior toform a major arc, MSA Major arcs are the "long way" aroundthe circle.ATMSMSmajor arcName another minor arc.Major arcs are greater than 180o. HighlightMSAMajor arcs are named by their endpoints and a point on thearc.Name another major arc.Slide 27 / 150. .HSTMeasurement By A Central Angleminor arcAnswerMSlide 28 / 150AThe measure of a minor arc is the measure of its central angle.The measure of the major arc is 3600 minus the measure of thecentral angle.B400A semicircle is an arc whose endpoints are theendpoints of the diameter.A4 00G0 32003600 - 40DMAT is a semicircle. Highlight the semicircle.Semicircles are named by their endpoints and a point onthe arc.Name another semicircle.Slide 29 / 150Slide 30 / 150The Length of the Arc Itself (AKA - Arc Length)EXAMPLEArc length is a portion of the circumference of a circle.In A , the central angle is 600 and the radius is 8 cm.Arc Length Corollary - In a circle, the ratio of the length ofa given arc to the circumference is equal to the ratio of themeasure of thearc to 3600.Find the length of CTAarc length of CTr CT3600orTCTarc length of CT3600.AnswerCC8 cmA6 00T

Slide 31 / 150Slide 32 / 150EXAMPLE10 In circle C whereIn A , the central angle is 400 and the length of SYis 4.19 in. Find the circumference of A.1 3 50D4 .1 9 inAnswerAB4 00C1 5 inAAnswerSis a diameter, findYSlide 33 / 15011 In circle C, whereSlide 34 / 15012 In circle C, whereis a diameter, findis a diameter, findBB1 3 50D1 3 50DAnswerC1 5 inAAnswerAC1 5 inSlide 35 / 150Slide 36 / 15014 Find the length ofA1 3 50DCAAnswerYesNoBB4 50C3 cmAnswer13 In circle C can it be assumed that AB is a diameter?

Slide 37 / 150Slide 38 / 15015 Find the circumference of circle T.16 In circle T, WY & XZ are diameters. WY XZ 6.If XY 1400 , what is the length of YZ?BTCAnswer6 .8 2 cmDZSlide 39 / 150EXAMPLEA result of a survey about the ages of people in a city are shown.TFind the indicated measures.S 651.Just as with adjacent angles, measures of adjacent arcs can beadded to find the measure of the arc formed by the adjacent arcs.3 002.9 0017-44.C.3. A45-64RV15-17Slide 41 / 150Slide 42 / 150Match the type of arc and it's measure to the given arcs below:TS16001800sem icircle24006 00RTeacher Notes8 00m ajor arcCONGRUENT CIRCLES & ARCS· Two circles are congruent if they have the same radius.· Two arcs are congruent if they have the same measure and theyare arcs of the same circle or congruent circles.TQ1 2 001200U8 006 004.m inor arc1 0 00AnswerAdjacent arcs: two arcs of the same circle are adjacent if theyhave a common endpoint.800YSlide 40 / 150ADJACENT ARCSTXWAnswerAT7 50DC5 50RE5 50Fbecause they are in thesame circle andSUhave the same&measure, but are not congruentbecause they are arcs of circlesthat are not congruent.

Slide 43 / 150Slide 44 / 150A17TrueFalseM18LBTrue7 001 8 00 4 00FalseC8 50PNAnswerAnswerDSlide 45 / 15019 Circle P has a radius of 3 andof 900 . What is the length of?Slide 46 / 15020 Two concentric circles always havecongruent radii.has a measureAABTrueFalseCDAnswerPAnswerBSlide 47 / 150Slide 48 / 15021 If two circles have the same center, they arecongruent.22 Tanny cuts a pie into 6 congruent pieces. What isthe measure of the central angle of each piece?AnswerFalseAnswerTrue

Slide 49 / 150Slide 50 / 150Click on the link below and complete thelabs before the Chords lesson.Chords, InscribedAngles & PolygonsLab - Properties of ChordsReturn to thetable ofcontentsSlide 51 / 150When a minor arc and a chord have the same endpoints, we callthe arc The Arc of the Chord.P.CQis the arc ofSlide 52 / 150THEOREM:In a circle, if one chord is a perpendicular bisector of another chord,then the first chord is a diameter.Tis the perpendicular bisector ofTherefore,**Recall the definition of a chord a segment with endpoints on thecircle.is a diameter of the circle.ACSEPTHEOREM:In the same circle, or in congruent circles, two minor arcs arecongruent if and only if their corresponding chords are congruent.BCiffTherefore,XEQSlide 54 / 150is a diameter of the circleand is perpendicular to chord.SLikewise, the perpendicularbisector of a chord of a circlepasses through the center of acircle.Slide 53 / 150THEOREM:If a diameter of a circle is perpendicular to a chord, then thediameter bisects the chord and its arc.AD*iff stands for "if and only if"

Slide 55 / 150Slide 56 / 150EXAMPLEBISECTING ARCSX., and(9 x)AAnswerYIf, then point Y and any linesegment, or ray, that contains Y,bisects,0DZ(80 - x) 0ESlide 57 / 150Slide 58 / 150C.GDEAEXAMPLEGiven circle C, QR ST 16.Find CU.RFUiffQand24 Given circle C below, the length of. FindC.AnswerBTSlide 60 / 150ARC5x - 9VSlide 59 / 15010802xSB23 In circle R,.Since the chords QR & ST arecongruent, they are equidistantfrom C. Therefore,AnswerTHEOREM:In the same circle, or congruent circles, two chords are congruent ifand only if they are equidistant from the center.A5B10C15D20is:ADB10.CDFAnswerCFind:CB

Slide 61 / 150Slide 62 / 15025 Given: circle P, PV PW, QR 2x 6, andST 3x - 1. Find the length of QR.B7C20D8True.QFalseRV3MSP35SWHTSlide 63 / 150Slide 64 / 150INSCRIBED ANGLESTHEOREM:The measure of an inscribed angle is half themeasure of its intercepted arc.DInscribed angles are angles whosevertices are in on the circle andwhose sides are chords of thecircle.The arc that lies in the interior ofan inscribed angle, and hasendpoints on the angle, is calledthe intercepted arc.OCGTis an inscribedangle andis its intercepted arc.AClick on the link below and complete the lab.Lab - Inscribed AnglesSlide 65 / 150Slide 66 / 150EXAMPLEandRQ.5 004 80PTTHEOREM:If two inscribed angles of a circle intercept the same arc,then the angles are congruent.SAAnswerFindTBDCsince they bothinterceptAnswer1AAnswerA26 AH is a diameter of the circle.

Slide 67 / 150Slide 68 / 150In a circle, parallel chords intercept congruent arcs.27 Given circle C below, findD.CDIn circle O, if., then3 50AO1 0 00AnswerCEBBSlide 69 / 150Slide 70 / 15028 Given circle C below, findRE.CBAnswer3 50AA1 0 00BCSUDTSlide 71 / 150Slide 72 / 15030 Find31 In a circle, two parallel chords on opposite sidesof the center have arcs which measure 1000and 1200. Find the measure of one of the arcsincluded between the chords.ZPAnswerXYAnswerD29 Given the figure below, which pairs of angles arecongruent?AnswerA

Slide 73 / 150Slide 74 / 15032 Given circle O, find the value of x.33 Given circle O, find the value of x.x1 0 00.CBAD3 50.OCAnswerOAnswerBA3 00DxSlide 75 / 150Slide 76 / 150Try ThisINSCRIBED POLYGONSandIn the circle below,, andA polygon is inscribed if all its vertices lie on a circle.Find.Q213T4.SSlide 77 / 150THEOREM:If a right triangle is inscribed in a circle, then thehypotenuse is a diameter of the circle.LGiff AC is a diameter of thecircle.THEOREM:A quadrilateral can be inscribed in a circle if and only if itsopposite angles are supplementary.E.NxinscribedquadrilateralSlide 78 / 150A.inscribedtriangleAnswerPCRN, E, A, and R lie on circle C iffA

Slide 79 / 150Slide 80 / 150EXAMPLE34 The value of x isK2b2aB 980AnswerLA 150MC6 8004b2aB xSlide 81 / 150AC 60036 What is the value of x?AB.DA5B10C13D15E(1 2 x 40 0 )(8 x 100 )GFAnswerD 1200CAnswerB 300Slide 82 / 150is a central angle. What is?A 150DyD 1800J35 In the diagram,and8 20C 1120AnswerFind the value of each variable:Slide 83 / 150Slide 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 isperpendicular to a radius of the circleat its endpoint on the circle (the point of tangency).Tangents & SecantsReturn to thetable ofcontentsLine l is tangent to circle X iff lB would be the point of tangency.B.XClick on the link below and complete the lab.Lab - Tangent Lines.l

Slide 85 / 150Slide 86 / 150Finding the Radius of a CircleVerify A Line is Tangent to a CircleGiven:Isis a radius of circle PIf B is a point of tangency, find the radiusof circle C.tangent to circle P?12.35A5 0 ftPAnswer}S37r8 0 ftrBSlide 87 / 150Slide 88 / 150EXAMPLETHEOREM:Tangent segments from a common external point are congruent.Given: RS is tangent to circle C at S and RT is tangent to circle Cat T. Find x.RSCTIf AR and AT are tangent segments,thenSlide 89 / 1506067AnswerA}.38 S is a point of tangency. Find the radius r ofcircle T.C253x 4Slide 90 / 15037 AB is a radius of circle A. Is BC tangent to circle A?BRAnswerTNo28.AA31.7B60TC14rD3.5.Sr3 6 cm4 8 cmRAnswer.PYes.CAnswerT

Slide 91 / 150Slide 92 / 15039 In circle C, DA is tangent at A and DB is tangent atB. Find x.40 AB, BC, and CA are tangents to circle O. AD 5,AC 8, and BE 4. Find the perimeter of triangleABC.BA25CFAnswer3x - 8B.OEDAnswer.CDSlide 93 / 150ASlide 94 / 150A Tangent and a ChordTHEOREM:If a tangent and a chord intersect at a point on a circle, then themeasure of each angle formed is one half the measure of itsintercepted arc.Tangents and secants can form other anglerelationships in circle. Recall the measure of aninscribed angle is 1/2 its intercepted arc. This canbe extended to any angle that has its vertex on thecircle. This includes angles formed by twosecants, a secant and a tangent, a tangent and achord, and two tangents.M.A.2 1RSlide 95 / 150Slide 96 / 150A Tangent and a Secant, Two Tangents, and Two SecantsTHEOREM:If a tangent and a secant, two tangents, or two secants intersectoutside a circle, then the measure of the angle formed is half thedifference of its intercepted arcs.a tangent and asecanttwo tangents.Q2C1XP1MAMtwo secantsBATHEOREM:If two chords intersect inside a circle, then the measure of eachangle is half the sum of the intercepted arcs by the angle andvertical angle.HW32ZYT

Slide 97 / 150Slide 98 / 150EXAMPLEEXAMPLEFind the value of x.Find the value of x.x07 601 7 80AnswerCBAAnswer1 3 00Dx01 5 60Slide 99 / 150Slide 100 / 15041 Find the value of x.42 Find the value of x.C(3 x - 02 )7 80(x 60 )FD3 40AnswerSlide 101 / 150Slide 102 / 15043 Find44 FindAnswerB6 50A2 6 001AnswerE4 20AnswerHx0

Slide 103 / 150Slide 104 / 150To find the angle, you need the measure of both intercepted arcs.First, find the measure of the minor arc. Then we cancalculate the measure of the angle x0.45 Find the value of x.4 50B2 4 70Answerx.05Answer122x0ASlide 105 / 150Slide 106 / 15046 Find the value of x.47 Find the value of x.Students type their answers herex0x0Slide 107 / 150Slide 108 / 15048 Find the value of x49 Find the value of x.Students type their answers hereAnswerStudents type their answers here5 00Answer1 0 001 2 00x0(5 x 100 )Answer2 2 00AnswerStudents type their answers here

Slide 109 / 150Slide 110 / 15050 Find the value of x.Segments & CirclesAnswer(2 x - 300 )xReturn to thetable ofcontentsSlide 111 / 150Slide 112 / 150EXAMPLETHEOREM:If two chords intersect inside a circle, then the products of themeasures of the segments of thechords are equal.Find the value of x.CA45E5xBSlide 113 / 150Slide 114 / 150EXAMPLE51 Find the value of x.Find ML & JK.xKx 2Lxx x 14AnswerM16918AnswerDJAnswer3 00

Slide 115 / 150Slide 116 / 150THEOREM:If two secant segments are drawn to a circle from an external point,then the product of the measures of one secant segment and itsexternal secant segment equals the product of the measures of theother secant segment and its external secant segment.52 Find the value of x.-2B4C5D6x2x 62AnswerABxAEDCSlide 117 / 150Slide 118 / 150EXAMPLE53 Find the value of x.Find the value of x.3x5x 2x 1Answer6Answer9x- 1Slide 119 / 150Slide 120 / 150THEOREM:If a tangent segment and a secant segment are drawn to a circlefrom an external point, then the square of the measure of thetangent segment is equal to the product of the measures of thesecant segment and its external secant segment.54 Find the value of x.5x 4AnswerA4x- 2ECD

Slide 121 / 150Slide 122 / 15055 Find the value of x.EXAMPLEFind RS.QxSAnswerRT8Slide 123 / 1501Answer316xSlide 124 / 15056 Find the value of x.12Answer24xEquations of aCircleReturn to thetable ofcontents(x, y)rxyLet (x, y) be any point on a circlewith center at the origin andradius, r. By the PythagoreanTheorem,Slide 126 / 150EXAMPLEWrite the equation of the circle.x2 y2 r24This is the equation of a circle withcenter at the origin.AnswerSlide 125 / 150

Slide 127 / 150Slide 128 / 150EXAMPLEFor circles whose center is not at the origin, we use thedistance formula to derive the equation.rWrite the standard equation of a circle withcenter (-2, 3) & radius 3.8.(x, y)Answer(h, k)This is the standard equation ofa circle.Slide 129 / 150Slide 130 / 150EXAMPLEEXAMPLEThe point (-5, 6) is on a circle with center (-1, 3). Write thestandard equation of the circle.The equation of a circle is (x - 4)2 (y 2)2 36. Graph the circle.AnswerWe know the center of the circle is (4, -2) and the radius is 6.Slide 131 / 150.First plot the center and move 6places in each direction.Then draw the circle.Slide 132 / 15058 What is the standard equation of the circle?57 What is the standard equation of the circle below?A (x - 4)2 (y - 3)2 81A x2 y2 400C (x 4)2 (y 3)2 81C (x 10)2 (y - 10)2 400D (x 4)2 (y 3)2 9Answer10AnswerB (x - 4)2 (y - 3)2 9B (x - 10)2 (y - 10)2 400D (x - 10)2 (y 10)2 400.

Slide 133 / 150Slide 134 / 15059 What is the center of (x - 4)2 (y - 2)2 64?(4,2)C(-4, -2)D(4, -2)AnswerBSlide 135 / 150Slide 136 / 15061 The standard equation of a circle is(x - 2)2 (y 1)2 16.What is the diameter of the circle?62 Which point does not lie on the circle describedby the equation (x 2)2 (y - 4)2 25?2A(-2, -1)B4B(1, 8)C8C(3, 4)D16D(0, 5)AnswerASlide 137 / 150Answer(0,0)AnswerA60 What is the radius of (x - 4)2 (y - 2)2 64?Slide 138 / 150A sector of a circle is the portion of the circle enclosed by tworadii and the arc that connects them.BArea of a SectorMajor SectorMinor SectorACReturn to thetable ofcontents

Slide 139 / 150Slide 140 / 150BAnswerA64 Which arc borders the major sector?AWAAnswer63 Which arc borders the minor sector?BXCBYZDSlide 141 / 150Slide 142 / 150Finding the Area of a SectorLets think about the formula.1. Use the formula:We want to find the area of part of the circle, so theformula for the area of a sector is the fraction of thecircle multiplied by the area of the circleAwhen θ is in degreesBAnswerThe area of a circle is found byr 3 4 50CWhen the central angle is in degrees, the fractionof the circle is out of the total 360 degrees.Slide 143 / 150Slide 144 / 150Example:65 Find the area of the minor sector of the circle. Roundyour answer to the nearest hundredth.Find the Area of the major sector.AAnswer8 cm6 00T5 .5 cm 3 00ATAnswerCC

Slide 145 / 150Slide 146 / 15066 Find the Area of the major sector for the circle.Round your answer to the nearest thousandth.67 What is the central angle for the major sector of thecircle?Answer1 2 cm1 5 cmA8 50TGSlide 147 / 150Slide 148 / 15068 Find the area of the major sector. Round to thenearest hundredth.69 The sum of the major and minor sectors' areas isequal to the total area of the circle.TrueAnswerC1 5 cmA1 2 00FalseAnswerAAnswerCC1 2 00GSlide 150 / 150Answer70 A group of 10 students orders pizza. They order 512" pizzas, that contain 8 slices each. If they split thepizzas equally, how many square inches of pizzadoes each student get?71 You have a circular sprinkler in your yard. Thesprinkler has a radius of 25 ft. How many square feetdoes the sprinkler water if it only rotates 120degrees?AnswerSlide 149 / 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 concentric circles. . no points Slide 16 / 150

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