Geometry Circles - Currituck County Schools
New Jersey Center for Teaching and LearningSlide 1 / 149Progressive Mathematics InitiativeThis 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.Click to go to website: www.njctl.orgSlide 2 / 149GeometryCircles2014-03-31www.njctl.orgTable of ContentsParts of a CircleAngles & ArcsChords, Inscribed Angles & PolygonsTangents & SecantsSegments & CirclesEquations of a CircleArea of a SectorSlide 3 / 149Click on a topic to goto that section
Slide 4 / 149Parts of a CircleReturn to thetable ofcontentsA circle is the set of all points in aplane that are a fixed distance froma given point in the plane calledthe center.Slide 5 / 149centerThe 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 6 / 149
Slide 7 / 149is a chord of circle ARA chord is a segment that has itsendpoints on the circle.ACTis the diameter of circleAA diameter is a chord that goesthrough the center of the circle.All diameters of a circle arecongruent.AnswerMWhat are the radii in this diagram?MSlide 7 (Answer) / 149is a chord of circle ARA chord is a segment that has itsendpoints on the circle.Ais the diameter of circleAA diameter is a chord that goesthrough the center of the circle.&All diameters of a circle arecongruent.AnswerCTWhat are the radii in this diagram?[This object is a pull tab]Slide 8 / 149The relationship between the diameter andradiustheTThe measure of the diameter, d, istwice the measure of the radius, r.MCTherefore,orIn . AIf, then what is the length ofwhat is the length ofAnswerA
Slide 8 (Answer) / 149The relationship between the diameter andradiustheTThe measure of the diameter, d, istwice the measure of the radius, r.MCTherefore,AnswerAIn . AIforAC 5TC 10, then what is the length ofwhat is the length of[This object is a pull tab]Slide 9 / 1491A diameter of a circle is the longest chord of thecircle.TrueAnswerFalseSlide 9 (Answer) / 149A diameter of a circle is the longest chord of thecircle.TrueFalseAnswer1True[This object is a pull tab]
Slide 10 / 1492A radius of a circle is a chord of a circle.TrueAnswerFalseSlide 10 (Answer) / 1492A radius of a circle is a chord of a circle.TrueAnswerFalseFalse[This object is a pull tab]Slide 11 / 149Two radii of a circle always equal the length of adiameter of a circle.TrueFalseAnswer3
Slide 11 (Answer) / 1493Two radii of a circle always equal the length of adiameter of a circle.TrueAnswerFalseTrue[This object is a pull tab]Slide 12 / 149If the radius of a circle measures 3.8 meters, whatis the measure of the diameter?4If the radius of a circle measures 3.8 meters, whatis the measure of the diameter?Answer4AnswerSlide 12 (Answer) / 1497.6 m[This object is a pull tab]
Slide 13 / 149How many diameters can be drawn in a circle?A1B2C4Dinfinitely manyAnswer5Slide 13 (Answer) / 149How many diameters can be drawn in a circle?A1B2C4Dinfinitely manyAnswer5D[This object is a pull tab]A secant of a circle is a line thatintersects the circle at two points.Aline l is a secant of this circle.BDEklA tangent is a line in the plane ofa circle that intersects the circleat exactly one point (the point oftangency).line k is a tangentD 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 14 / 149
Slide 15 / 149COPLANAR CIRCLES are two circles in the same plane whichintersect at 2 points, 1 point, or no points.Coplanar circles that intersects in 1 point are called tangentcircles. Coplanar circles that have a common center are calledconcentric.2 points.tangentcircles1 pointconcentriccirclesno pointsSlide 16 / 149A Common Tangent is a line, ray, or segment that is tangent to 2coplanar circles.Internally tangent(tangent linepassesbetween them)Slide 17 / 149How many common tangent lines do the circleshave?Answer6Externally tangent(tangent line doesnot pass betweenthem)
Slide 17 (Answer) / 149How many common tangent lines do the circleshave?Answer64[This object is a pull tab]How many common tangent lines do the circleshave?7How many common tangent lines do the circleshave?Slide 18 / 149AnswerAnswer71[This object is a pull tab]Slide 18 (Answer) / 149
How many common tangent lines do the circleshave?8How many common tangent lines do the circleshave?Slide 19 / 149AnswerAnswer8Slide 19 (Answer) / 1492[This object is a pull tab]Slide 20 / 149How many common tangent lines do the circleshave?Answer9
Slide 20 (Answer) / 149How many common tangent lines do the circleshave?Answer90[This object is a pull tab]Slide 21 / 149Using the diagram below, match the notation with the term thatbest describes tertangentcommon tangentpoint of tangencyUsing the diagram below, match the notation with the term thatbest describes mmon TangentChordSecantTangentPoint of TangencyRadiusDiameterdiametertangent[This object is tangenta pull tab]commonpoint of tangency.F.GSlide 21 (Answer) / 149
Slide 22 / 149Angles & ArcsReturn to thetable ofcontentsSlide 23 / 149An 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.)Slide 24 / 149A central angle is an angle whose vertex is thecenter of the circle. .HSTAIn,is the centralangle.Name another central angle.AnswerM
Slide 24 (Answer) / 149A central angle is an angle whose vertex is thecenter of the circle. .HSATIn,AnswerMis the centralangle.Name another central angle.[This object is a pull tab]Slide 25 / 149Ifis less than 1800, then the points onthat lie in the interior ofform the minor arc withendpoints M and H. .HSminor arc MAAnswerMHighlight MAATName another minor arc.M. .HSTminor arc MAAnswerIfis less than 1800, then the points onthat lie in the interior ofform the minor arc withendpoints M and H.Highlight MAA[This object is a pull tab]Name another minor arc.Slide 25 (Answer) / 149
Slide 26 / 149major arcM. .AnswerHSATPoints M and A and all points ofexterior toform a major arc, MSA Major arcs are the "long way" aroundthe circle.Major arcs are greater than 180o. HighlightMSAMajor arcs are named by their endpoints and a point on thearc.Name another major arc.Slide 26 (Answer) / 149major arcM. .AnswerHSAT[This object is a pull tab]Points M and A and all points ofexterior toform a major arc, MSA Major arcs are the "long way" aroundthe circle.Major arcs are greater than 180o. HighlightMSAMajor arcs are named by their endpoints and a point on thearc.Name another major arc.Slide 27 / 149. .HSTminor arcAAnswerMA semicircle is an arc whose endpoints are theendpoints of the diameter.MAT is a semicircle. Highlight the semicircle.Semicircles are named by their endpoints and a point onthe arc.Name another semicircle.
Slide 27 (Answer) / 149. .HSminor arcAnswerMAT[This object is a pull tab]A semicircle is an arc whose endpoints are theendpoints of the diameter.MAT is a semicircle. Highlight the semicircle.Semicircles are named by their endpoints and a point onthe arc.Name another semicircle.Slide 28 / 149Measurement By A Central AngleThe 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.040G0 32003600 - 40DSlide 29 / 149The 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 ofa given arc to the circumference is equal to the ratio of themeasure of thearc to 3600.CAarc length of CTr CT3600orTCTarc length of CT3600.
Slide 30 / 149EXAMPLEIn A , the central angle is 600 and the radius is 8 cm.Find the length of CTAnswerC8 cmA6 00TSlide 30 (Answer) / 149EXAMPLEIn A , the central angle is 600 and the radius is 8 cm.Find the length of CT8 cmA6 00arc length ofCTAnswerC CT3600 3600600.8.38 cmT[This object is a pull tab]Slide 31 / 149EXAMPLEIn A , the central angle is 400 and the length of SYis 4.19 in. Find the circumference of A.A4 .1 9 inAnswerS4 00Y
Slide 31 (Answer) / 149EXAMPLEIn A , the central angle is 40 and the length of SYis 4.19 in. Find the circumference of A.0Aarc length of SY4 .1 9 inAnswerS4 00 SY36004.19 40036004.19 19Y 37.71 in[This object is a pull tab]10 In circle C whereSlide 32 / 149is a diameter, findB1 3 50DC1 5 in10 In circle C whereAnswerAis a diameter, findB1 3 50DAAnswerC1 5 in[This object is a pull tab]Slide 32 (Answer) / 149
11 In circle C, whereSlide 33 / 149is a diameter, findB0135DAnswerC1 5 inA11 In circle C, whereSlide 33 (Answer) / 149is a diameter, findB1 3 50DAnswerC1 5 inA[This object is a pull tab]12 In circle C, whereSlide 34 / 149is a diameter, findB1 3 50DC1 5 inAnswerA
12 In circle C, whereSlide 34 (Answer) / 149is a diameter, findB1 3 50DC1 5 inAnswerA[This object is a pull tab]Slide 35 / 14913 In circle C can it be assumed that AB is a diameter?NoAnswerBYes1 3 50DCA13 In circle C can it be assumed that AB is a diameter?No1 3 50DCAnswerBYesYes[This object is a pull tab]ASlide 35 (Answer) / 149
Slide 36 / 14914 Find the length ofAB4 503 cmAnswerCSlide 36 (Answer) / 14914 Find the length ofAB4 503 cmAnswerC[This object is a pull tab]Slide 37 / 14915 Find the circumference of circle T.T6 .8 2 cmAnswer7 50
Slide 37 (Answer) / 14915 Find the circumference of circle T.T7 50Answer6 .8 2 cm[This object is a pull tab]Slide 38 / 14916 In circle T, WY & XZ are diameters. WY XZ 6.If XY 1400 , what is the length of YZ?XWAnswerABTCDYZSlide 38 (Answer) / 14916 In circle T, WY & XZ are diameters. WY XZ 6.If XY 1400 , what is the length of YZ?XWAnswerABCDTZAY[This object is a pull tab]
Slide 39 / 149ADJACENT ARCSAdjacent arcs: two arcs of the same circle are adjacent if theyhave a common endpoint.Just as with adjacent angles, measures of adjacent arcs can beadded to find the measure of the arc formed by the adjacent arcs.CT. A.Slide 40 / 149EXAMPLEA result of a survey about the ages of people in a city are shown.TFind the indicated measures.S 651.9 0017-441 0 003.U8 006 004.Answer3 002.45-64RV15-17EXAMPLEA result of a survey about the ages of people in a city are shown.TFind the indicated measures.S 651.003000 60 80 1403.4.Answer2.17-44 09 00U00 000 1100 301308 006 000 800 900 2300 60000 360- 90 270[This object is a pull tab]45-64R15-17VSlide 40 (Answer) / 149
Slide 41 / 149Match the type of arc and it's measure to the given arcs below:TQ8 006 00RSm inor arcm ajor arc12008001600Teacher Notes1 2 00sem icircle18002400Match the type of arc and it's measure to the given arcs below:Teacher NotesTSlide 41 (Answer) / 149Q1 2 008 006 00Arc labels and measurements inthe box are infinitely cloned sothey can be pulled up andmatched with the arc.SR[This object is a pull tab]m inor arcm ajor arc120080016001800sem icircle2400CONGRUENT 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.TDC5 50RE5 50Fbecause they are in thesame circle andSUhave the same&measure, but are not congruentbecause they are arcs of circlesthat are not congruent.Slide 42 / 149
Slide 43 / 149A17BTrueFalse7 004 001 8 00CAnswerDSlide 43 (Answer) / 149A17TrueFalseB7 001 8 00 4 00CDAnswerTrue[This object is a pull tab]Slide 44 / 149M18LTrue8 50PNAnswerFalse
Slide 44 (Answer) / 149M18LTrue8 50FalsePNAnswerFalse[This object is a pull tab]19 Circle P has a radius of 3 andof 900 . What is the length ofSlide 45 / 149?AABChas a measurePDAnswerB19 Circle P has a radius of 3 andof 900 . What is the length ofDPBAnswerC?AABSlide 45 (Answer) / 149has a measureA[This object is a pull tab]
Slide 46 / 14920 Two concentric circles always havecongruent radii.TrueAnswerFalseSlide 46 (Answer) / 14920 Two concentric circles always havecongruent radii.TrueAnswerFalseFalse[This object is a pull tab]Slide 47 / 14921 If two circles have the same center, they arecongruent.TrueAnswerFalse
Slide 47 (Answer) / 14921 If two circles have the same center, they arecongruent.TrueAnswerFalseFalse[This object is a pull tab]Slide 48 / 149Answer22 Tanny cuts a pie into 6 congruent pieces. What isthe measure of the central angle of each piece?Answer22 Tanny cuts a pie into 6 congruent pieces. What isthe measure of the central angle of each piece?[This object is a pull tab]Slide 48 (Answer) / 149
Slide 49 / 149Chords, InscribedAngles & PolygonsReturn to thetable ofcontentsSlide 50 / 149When a minor arc and a chord have the same endpoints, we callthe arc The Arc of the Chord.P.Cis the arc ofQ**Recall the definition of a chord a segment with endpoints on thecircle.Slide 51 / 149THEOREM: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,.is a diameter of the circle.Likewise, the perpendicularbisector of a chord of a circlepasses through the center of acircle.SEQP
THEOREM:If a diameter of a circle is perpendicular to a chord, then thediameter bisects the chord and its arc.ACSlide 52 / 149is a diameter of the circleand is perpendicular to chord.Therefore,XSETHEOREM:In the same circle, or in congruent circles, two minor arcs arecongruent if and only if their corresponding chords are congruent.BSlide 53 / 149CiffDA*iff stands for "if and only if"BISECTING ARCSXCYZIf, then point Y and any linesegment, or ray, that contains Y,bisectsSlide 54 / 149
Slide 55 / 149EXAMPLEFind:CB.,, and(9 x)0AnswerAD(80 - x) 0ESlide 55 (Answer) / 149EXAMPLEFind:CB.,(9 x)0AnswerADE(80 - x) 0, and9x 80 - x10x 80x 8 9(8) 072 80 - 8 0 72000 72 72 144[This object is a pull tab]THEOREM:In the same circle, or congruent circles, two chords are congruent ifand only if they are equidistant from the center.CA.GDEFBiffSlide 56 / 149
Slide 57 / 149EXAMPLEGiven circle C, QR ST 16.Find CU.RQ.2xCT5x - 9VAnswerUSince the chords QR & ST arecongruent, they are equidistantfrom C. Therefore,SSlide 57 (Answer) / 149EXAMPLEGiven circle C, QR ST 16.Find CU.RQ.2xC5x - 9VSAnswerUSince the chords QR & ST arecongruent, they are equidistantfrom C. Therefore,2x T 5x - 99 3x3 xCU 2(3) 6[This object is a pull tab]Slide 58 / 14923 In circle R,and. FindA1080C.AnswerRBD
Slide 58 (Answer) / 14923 In circle R,and. FindAC1080.AnswerR1080BD[This object is a pull tab]Slide 59 / 14924 Given circle C below, the length of5B10C15D20ADB10.CAnswerAis:FSlide 59 (Answer) / 14924 Given circle C below, the length of5B10C15D20ADAnswerAis:B10.DC[This object is a pull tab]F
Slide 60 / 14925 Given: circle P, PV PW, QR 2x 6, andST 3x - 1. Find the length of QR.1B7C20D8RV.QSPAnswerAWTSlide 60 (Answer) / 14925 Given: circle P, PV PW, QR 2x 6, andST 3x - 1. Find the length of QR.B7C20D8.QPTSCW[This object is a pull tab]Slide 61 / 14926 AH is a diameter of the circle.ATrueFalseRV3M3SH5TAnswer1AnswerA
Slide 61 (Answer) / 14926 AH is a diameter of the circle.ATrue3FalseM5TSAnswer3HFalse[This object is a pull tab]Slide 62 / 149INSCRIBED ANGLESDInscribed 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.Ois an inscribedangle andis its intercepted arc.THEOREM:The measure of an inscribed angle is half themeasure of its intercepted arc.CTAGSlide 63 / 149
Slide 64 / 149EXAMPLEFindand.AnswerRQ5 004 80PSTSlide 64 (Answer) / 149EXAMPLEandRQ.5 004 80PAnswerFindST[This object is a pull tab]THEOREM:If two inscribed angles of a circle intercept the same arc,then the angles are congruent.ABDCsince they bothinterceptSlide 65 / 149
Slide 66 / 149In a circle, parallel chords intercept congruent arcs.CDIn circle O, if., thenOBASlide 67 / 14927 Given circle C below, findDE.CAAnswer3 501 0 00BSlide 67 (Answer) / 14927 Given circle C below, findD.CA3 501 0 00AnswerE500B[This object is a pull tab]
Slide 68 / 14928 Given circle C below, findDE.CAnswer3 50A1 0 00BSlide 68 (Answer) / 14928 Given circle C below, findDE.CABAnswer3 501 0 001100[This object is a pull tab]Slide 69 / 14929 Given the figure below, which pairs of angles arecongruent?RBCSUDTAnswerA
Slide 69 (Answer) / 14929 Given the figure below, which pairs of angles arecongruent?RASCBUAnswerDAT[This object is a pull tab]Slide 70 / 14930 FindX.PAnswerYZSlide 70 (Answer) / 14930 FindY.ZPAnswerX900[This object is a pull tab]
Slide 71 / 149Answer31 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.Slide 71 (Answer) / 149Answer31 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.700[This object is a pull tab]Slide 72 / 14932 Given circle O, find the value of x.xBC.ODAnswerA3 00
Slide 72 (Answer) / 14932 Given circle O, find the value of x.xBA.3 00CDAnswerO1200[This object is a pull tab]Slide 73 / 14933 Given circle O, find the value of x.1 0 00BA.OCAnswer3 50DxSlide 73 (Answer) / 14933 Given circle O, find the value of x.1 0 00BA.OCAnswer3 501200Dx[This object is a pull tab]
Slide 74 / 149Try ThisandIn the circle below,, andFindQ2P1TAnswer34SSlide 74 (Answer) / 149Try ThisandIn the circle below,, andFindQ2P1TAnswer34S[This object is a pull tab]Slide 75 / 149INSCRIBED POLYGONSA polygon is inscribed if all its vertices lie on a circle.inscribedtriangle.inscribedquadrilateral.
Slide 76 / 149THEOREM:If a right triangle is inscribed in a circle, then thehypotenuse is a diameter of the circle.A.xiff AC is a diameter of thecircle.LGSlide 77 / 149THEOREM:A quadrilateral can be inscribed in a circle if and only if itsopposite angles are supplementary.E.NN, E, A, and R lie on circle C iffCARSlide 78 / 149EXAMPLEFind the value of each variable:K2b2a4b2aJMAnswerL
Slide 78 (Answer) / 149EXAMPLEFind the value of each variable:K2b2a4b2aMAnswerL2a 2a 1804a 180a 4504b 2b 1806b 180b 300J[This object is a pull tab]Slide 79 / 14934 The value of x isC6 80A 1500B 980B x8 20C 1120DyD 1800AnswerASlide 79 (Answer) / 14934 The value of x isA 150C6 800B 980B x8 20C 1120yAAnswerD 1800DB[This object is a pull tab]
35 In the diagram,andSlide 80 / 149is a central angle. What is?A 150ABC 600.CDAnswerB 300D 120035 In the diagram,andSlide 80 (Answer) / 149is a central angle. What is?A 150B 300ABC 600.CDBAnswerD 1200[This object is a pull tab]Slide 81 / 14936 What is the value of x?5B10C13D15(1 2 x 40 0 )(8 x 100 )FGAnswerAE
Slide 81 (Answer) / 14936 What is the value of x?5B10C13D15(1 2 x 40 0 )(8 x 100 )GFAnswerAEA[This object is a pull tab]Slide 82 / 149Tangents & SecantsReturn to thetable ofcontentsSlide 83 / 149**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).Line l is tangent to circle X iff lB would be the point of tangency.B.Xl
Slide 84 / 149Verify A Line is Tangent to a CircleGiven:TIsis a radius of circle Ptangent to circle P?12.35PAnswer}S37Slide 84 (Answer) / 149Verify A Line is Tangent to a CircleGiven:TIsis a radius of circle Ptangent to circle P?12.35Answer}SP37Since 352 122 372, triangle PSTis a right triangle. Therefore, ST isperpendicular to radius TP at itsendpoint on circle P. So, ST istangent to circle P at T.[This object is a pull tab]Slide 85 / 149Finding the Radius of a CircleA5 0 ftr8 0 ft.CrBAnswerIf B is a point of tangency, find the radiusof circle C.
Slide 85 (Answer) / 149Finding the Radius of a CircleIf B is a point of tangency, find the radiusof circle C.5 0 ftr8 0 ft.CrAC2 BC2 AB2802 r2 (50 r)26400 r2 r2 100r 25006400 100r 25003900 100r39 rSo, r 39 ft.AnswerAB[This object is a pull tab]Slide 86 / 149THEOREM:Tangent segments from a common external point are congruent.R.PATIf AR and AT are tangent segments,thenSlide 87 / 149EXAMPLEGiven: RS is tangent to circle C at S and RT is tangent to circle Cat T. Find x.SRT3x 4Answer28.C
Slide 87 (Answer) / 149EXAMPLEGiven: RS is tangent to circle C at S and RT is tangent to circle Cat T. Find x.S28RT3x 43x 4 283x 24x 8Answer.C[This object is a pull tab]Slide 88 / 14937 AB is a radius of circle A. Is BC tangent to circle A?NoB60C25}.AAnswerYes6737 AB is a radius of circle A. Is BC tangent to circle A?NoB25C}.A60AnswerYes67No[This object is a pull tab]Slide 88 (Answer) / 149
Slide 89 / 14938 S is a point of tangency. Find the radius r ofcircle T.31.7B60TC14rD3.5.r3 6 cmSR4 8 cmAnswerASlide 89 (Answer) / 14938 S is a point of tangency. Find the radius r ofcircle T.A31.7B60TC14rD3.5.3 6 cmR4 8 cmAnswerSrC[This object is a pull tab]Slide 90 / 14939 In circle C, DA is tangent at A and DB is tangent atB. Find x.A25.CB3x - 8AnswerD
Slide 90 (Answer) / 14939 In circle C, DA is tangent at A and DB is tangent atB. Find x.A25.CAnswerD3x - 8B[This object is a pull tab]Slide 91 / 14940 AB, BC, and CA are tangents to circle O. AD 5,AC 8, and BE 4. Find the perimeter of triangleABC.BCAnswerFE.OADSlide 91 (Answer) / 149FEC.ODAnswer40 AB, BC, and CA are tangents to circle O. AD 5,AC 8, and BE 4. Find the perimeter of triangleABC.BA[This object is a pull tab]
Slide 92 / 149Tangents 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.Slide 93 / 149A 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.M.A.2 1RSlide 94 / 149A 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 tangentstwo secantsBXPA1.Q2CMW3ZY
Slide 95 / 149THEOREM: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.AM12HTSlide 96 / 149EXAMPLEFind the value of x.Dx07 601 7 80AnswerCBASlide 96 (Answer) / 149EXAMPLEFind the value of x.Cx07 601 7 80BAAnswerD[This object is a pull tab]
Slide 97 / 149EXAMPLEFind the value of x.Answer1 3 00x01 5 60Slide 97 (Answer) / 149EXAMPLEFind the value of x.Answer1 3 000xx 1/2 (1300 1560)x 1430[This object is a pull tab]1 5 60Slide 98 / 14941 Find the value of x.C7 80E4 20DFAnswerHx0
Slide 98 (Answer) / 14941 Find the value of x.C7 80E4 20FDAnswerHx0[This object is a pull tab]Slide 99 / 14942 Find the value of x.(3 x - 02 )(x 60 )Answer3 40Slide 99 (Answer) / 14942 Find the value of x.(3 x - 02 )(x 60 )Answer3 40[This object is a pull tab]
Slide 100 / 14943 FindAnswerB6 50ASlide 100 (Answer) / 14943 FindAnswerB6 50[This object is a pull tab]ASlide 101 / 1492 6 001Answer44 Find
Slide 101 (Answer) / 149Answer44 Find12 6 00[This object is a pull tab]Slide 102 / 14945 Find the value of x.05Answerx1224 50Slide 102 (Answer) / 14945 Find the value of x.05Answerx1224 50[This object is a pull tab]
Slide 103 / 149To 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.AnswerB2 4 70x0ASlide 103 (Answer) / 149To 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.First find the minor arc.AnswerB2 4 70x0[This object is a pull tab]ASlide 104 / 14946 Find the value of x.2 2 00x0AnswerStudents type their answers here
Slide 104 (Answer) / 14946 Find the value of x.Students type their answers hereAnswerFirst find the minor arc.2 2 00x0[This object is a pull tab]Slide 105 / 14947 Find the value of x.x0AnswerStudents type their answers here1 0 00Slide 105 (Answer) / 14947 Find the value of x.Students type their answers herex01 0 00AnswerFirst find the major arc.[This object is a pull tab]
Slide 106 / 14948 Find the value of xAnswerStudents type their answers here5 00x0Slide 106 (Answer) / 14948 Find the value of xStudents type their answers hereAnswerFind the major arc.5 00x0[This object is a pull tab]Slide 107 / 14949 Find the value of x.1 2 00(5 x 100 )AnswerStudents type their answers here
Slide 107 (Answer) / 14949 Find the value of x.Students type their answers hereAnswerFind the major arc.1 2 00(5 x 100 )[This object is a pull tab]Slide 108 / 149(2 x - 300 )3 00xSlide 108 (Answer) / 149Answer50 Find the value of x.3 00Answer50 Find the value of x.x(2 x - 300 )[This object is a pull tab]
Slide 109 / 149Segments & CirclesReturn to thetable ofcontentsSlide 110 / 149THEOREM:If two chords intersect inside a circle, then the products of themeasures of the segments of thechords are equal.CAEDBSlide 111 / 149EXAMPLE455xAnswerFind the value of x.
Slide 111 (Answer) / 149EXAMPLEAnswerFind the value of x.455x[This object is a pull tab]Slide 112 / 149EXAMPLEFind ML & JK.MKLxx 4Answerx 2x 1JEXAMPLEFind ML & JK.AnswerMKx 2Lx 4x ML x (21 2) ( 2 1) 7JJK 2 (2 4) 8[This object is a pull tab]Slide 112 (Answer) / 149
Slide 113 / 14951 Find the value of x.x9Answer1618Slide 113 (Answer) / 14951 Find the value of x.16918Answerx[This object is a pull tab]Slide 114 / 14952 Find the value of x.-2B4C5D6x2x 62xAnswerA
Slide 114 (Answer) / 14952 Find the value of x.-2B4C5DxAnswerA2x 626xD[This object is a pull tab]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 itsexternal secant segment equals the product of the measures of theother secant segment and its external secant segment.Slide 115 / 149BAEDCSlide 116 / 149EXAMPLE96x5AnswerFind the value of x.
Slide 116 (Answer) / 149EXAMPLE96x5AnswerFind the value of x.[This object is a pull tab]Slide 117 / 14953 Find the value of x.3x 2Answerx 1x- 1Slide 117 (Answer) / 14953 Find the value of x.3x 2Answerx 1x- 1[This object is a pull tab]
Slide 118 / 14954 Find the value of x.5Answer4x 4x- 2Slide 118 (Answer) / 14954 Find the value of x.5Answer4x 4x- 2[This object is a pull tab]THEOREM: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.AECDSlide 119 / 149
Slide 120 / 149EXAMPLEFind RS.QRxSAnswer16T8Slide 120 (Answer) / 149EXAMPLEFind RS.QRAnswer16xST8Since we are dealing withmeasurement, we only wantthe positive answer:[This object is a pull tab]Slide 121 / 14931xAnswer55 Find the value of x.
Slide 121 (Answer) / 14955 Find the value of x.Answer31x[This object is a pull tab]Slide 122 / 14956 Find the value of x.12Answer24xSlide 122 (Answer) / 14956 Find the value of x.12xAnswer24[This object is a pull tab]
Slide 123 / 149Equations of aCircleReturn to thetable ofcontentsSlide 124 / 149(x, y)ryLet (x, y) be any point on a circlewith center at the origin andradius, r. By the PythagoreanTheorem,xx2 y2 r2This is the equation of a circle withcenter at the origin.Slide 125 / 149EXAMPLE4AnswerWrite the equation of the circle.
Slide 125 (Answer) / 149EXAMPLEAnswerWrite the equation of the circle.4x2 y2 (4)2x2 y2 16[This object is a pull tab]Slide 126 / 149For circles whose center is not at the origin, we use thedistance formula to derive the equation.r(x, y).(h, k)This is the standard equation ofa circle.Slide 127 / 149EXAMPLEAnswerWrite the standard equation of a circle withcenter (-2, 3) & radius 3.8.
Slide 127 (Answer) / 149EXAMPLEAnswerWrite the standard equation of a circle withcenter (-2, 3) & radius 3.8.[This object is a pull tab]Slide 128 / 149EXAMPLEAnswerThe point (-5, 6) is on a circle with center (-1, 3). Write thestandard equation of the circle.EXAMPLEThe point (-5, 6) is on a circle with center (-1, 3). Write thestandard equation of the circle.AnswerFirst, we need to find the length ofthe radius:Then substitute the center andradius into the standard equationof a circle:[This object is a pull tab]Slide 128 (Answer) / 149
Slide 129 / 149EXAMPLEThe 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 6places in each direction.Then draw the circle.Slide 130 / 14957 What is the standard equation of the circle below?A x2 y2 400B (x - 10)2 (y - 10)2 400C (x 10)2 (y - 10)2 40010AnswerD (x - 10)2 (y 10)2 40057 What is the standard
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 15 / 149
Two circles are congruent circles if and only if they have congruent radii. A B if AC BD . AC BD if A B. Concentric circles are coplanar circles with the same center. Two coplanar circles that intersect at exactly one point are called tangent circles . Internally tangent circles Externally tangent circles Pairs of Circles
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
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 15 / 150
Geometry Unit 10: Circles Name_ Geometry Unit 10: Circles Ms. Talhami 2 Helpful Vocabulary Word Definition/Explanation Examples/Helpful Tips Geometry Unit 10: Circles Ms. Talhami 3 Equation of a Circle Determine the center an
Coplanar Circles and Common Tangents In a plane, two circles can intersect in two points, one point, or no points. Coplanar circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric circles. A line or segment that is tangent to two coplanar circles is called a common tangent. A
circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric circles. A line or segment that is tangent to two coplanar circles is called a common internal tangent A common external tangent two circles. Notes: intersects the segment that joins the centers of the two circles .
Senior Formal Pictures During the summer, Currituck County High School has Lifetouch Studios come out to the school to take senior formals. They can also take casuals if desired. . * Senior ads will go on sale the first day of the school year and will be discounted until Nov 10, 2017. * All senior ads, ad layouts and order forms must be .
A CENSUS LIST OF WOOL ALIENS FOUND IN BRITAIN, 1946-1960 221 A CENSUS LIST OF WOOL ALIENS FOUND IN BRITAIN, 1946-1960 Compiled by J. E. LOUSLEY Plants introduced into Britain by the woollen industry have attracted increasing interest from field botanists in recent years and this follows a long period of neglect. Early in the present century Ida M. Hayward, assisted by G. C. Druce, made a .
