Parts Of A Circle Geometry Circles

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