Geometry Circles - NJCTL
New Jersey Center for Teaching and LearningSlide 1 / 150Progressive 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.Click to go to website:www.njctl.orgSlide 2 / 150GeometryCircles2014-06-03www.njctl.orgTable of ContentsParts of a CircleAngles & ArcsChords, Inscribed Angles & PolygonsTangents & SecantsSegments & CirclesEquations of a CircleArea of a SectorSlide 3 / 150Click on a topic to goto that section
Slide 4 / 150Parts 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 / 150centerThe 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 / 150
Slide 7 / 150is 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?Slide 8 / 150The 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 ofAnswerAwhat is the length ofSlide 9 / 150A diameter of a circle is the longest chord of thecircle.TrueFalseAnswer1
Slide 10 / 1502A radius of a circle is a chord of a circle.TrueAnswerFalseSlide 11 / 1503Two radii of a circle always equal the length of adiameter of a circle.TrueAnswerFalseSlide 12 / 150If the radius of a circle measures 3.8 meters, whatis the measure of the diameter?Answer4
Slide 13 / 150How many diameters can be drawn in a circle?A1B2C4Dinfinitely manyAnswer5Slide 14 / 150A secant of a circle is a line thatintersects the circle at two points.Aline l is a secant of this circle.BlDEA 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.COPLANAR 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 15 / 150
Slide 16 / 150A Common Tangent is a line, ray, or segment that is tangent to 2coplanar circles.Internally tangent(tangent linepassesbetween them)Externally tangent(tangent line doesnot pass betweenthem)How many common tangent lines do the circleshave?7How many common tangent lines do the circleshave?Slide 17 / 150Answer6AnswerSlide 18 / 150
How many common tangent lines do the circleshave?9How many common tangent lines do the circleshave?Slide 19 / 150Answer8AnswerSlide 20 / 150Slide 21 / 150Using the diagram below, match the notation with the term thatbest describes tertangentcommon tangentpoint of tangency
Slide 22 / 150Angles & ArcsReturn to thetable ofcontentsSlide 23 / 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.)Slide 24 / 150A central angle is an angle whose vertex is thecenter of the circle. .HSTAIn,is the centralangle.Name another central angle.AnswerM
Slide 25 / 150Ifis less than 1800, then the points onthat lie in the interior ofform the minor arc withendpoints M and H.M. .HSAnswerminor arc MAHighlight MAATName another minor arc.Slide 26 / 150major 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 27 / 150. .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 28 / 150Measurement 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.4 00G0 32003600 - 40DSlide 29 / 150The 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 / 150EXAMPLEIn A , the central angle is 600 and the radius is 8 cm.Find the length of CTAnswerC8 cmA6 00T
Slide 31 / 150EXAMPLEIn A , the central angle is 40 and the length of SYis 4.19 in. Find the circumference of A.0SAnswerA4 .1 9 in4 00Y10 In circle C whereSlide 32 / 150is a diameter, findB1 3 50DC1 5 inAnswerA11 In circle C, whereSlide 33 / 150is a diameter, findBC1 5 inAAnswer1 3 50D
12 In circle C, whereSlide 34 / 150is a diameter, findB1 3 50DC1 5 inAnswerASlide 35 / 15013 In circle C can it be assumed that AB is a diameter?No1 3 50DCAnswerBYesASlide 36 / 15014 Find the length ofABC3 cmAnswer4 50
Slide 37 / 15015 Find the circumference of circle T.T7 50Answer6 .8 2 cmSlide 38 / 15016 In circle T, WY & XZ are diameters. WY XZ 6.If XY 1400 , what is the length of YZ?XWAnswerABTCDZYSlide 39 / 150ADJACENT 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 / 150EXAMPLEA result of a survey about the ages of people in a city are shown.TFind the indicated measures.S 651.3 009 0017-441 0 003.UAnswer2.8 006 004.45-64RV15-17Slide 41 / 150Match the type of arc and it's measure to the given arcs below:T8 006 00RSm inor arcm ajor arc120080016001800Teacher NotesQ1 2 00sem 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 / 150
Slide 43 / 150A17BTrueFalse1 8 007 004 00CAnswerDSlide 44 / 150M18LTrue8 50FalsePAnswerN19 Circle P has a radius of 3 andof 900 . What is the length ofDPBAnswerC?AABSlide 45 / 150has a measure
Slide 46 / 15020 Two concentric circles always havecongruent radii.TrueAnswerFalseSlide 47 / 15021 If two circles have the same center, they arecongruent.TrueAnswerFalseSlide 48 / 150Answer22 Tanny cuts a pie into 6 congruent pieces. What isthe measure of the central angle of each piece?
Slide 49 / 150Chords, InscribedAngles & PolygonsReturn to thetable ofcontentsSlide 50 / 150Click on the link below and complete thelabs before the Chords lesson.Lab - Properties of ChordsWhen a minor arc and a chord have the same endpoints, we callthe arc The Arc of the Chord.P.CQ**Recall the definition of a chord a segment with endpoints on thecircle.is the arc ofSlide 51 / 150
Slide 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,.is a diameter of the circle.SEQPLikewise, the perpendicularbisector of a chord of a circlepasses through the center of acircle.THEOREM:If a diameter of a circle is perpendicular to a chord, then thediameter bisects the chord and its arc.ACSlide 53 / 150is 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.BCiffAD*iff stands for "if and only if"Slide 54 / 150
Slide 55 / 150BISECTING ARCSXCYIf, then point Y and any linesegment, or ray, that contains Y,bisectsZSlide 56 / 150EXAMPLEFind:CB.,, and(9 x)0AnswerADE(80 - x) 0THEOREM:In the same circle, or congruent circles, two chords are congruent ifand only if they are equidistant from the center.CA.GDEFBiffSlide 57 / 150
Slide 58 / 150EXAMPLEGiven circle C, QR ST 16.Find CU.RQ.2xCT5x - 9VAnswerUSince the chords QR & ST arecongruent, they are equidistantfrom C. Therefore,SSlide 59 / 15023 In circle R,and. FindAC1080.AnswerRBDSlide 60 / 15024 Given circle C below, the length of5B10C15D20ADB10.CFAnswerAis:
Slide 61 / 15025 Given: circle P, PV PW, QR 2x 6, andST 3x - 1. Find the length of QR.1B7C20D8RV.QSPAnswerAWTSlide 62 / 15026 AH is a diameter of the circle.A3FalseM35TSAnswerTrueHSlide 63 / 150INSCRIBED 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.OGis an inscribedangle andis its intercepted arc.Click on the link below and complete the lab.Lab - Inscribed Angles
Slide 64 / 150THEOREM:The measure of an inscribed angle is half themeasure of its intercepted arc.CTASlide 65 / 150EXAMPLEFindand.AnswerRQ5 004 80PSTTHEOREM:If two inscribed angles of a circle intercept the same arc,then the angles are congruent.ABDCsince they bothinterceptSlide 66 / 150
Slide 67 / 150In a circle, parallel chords intercept congruent arcs.CDIn circle O, if., thenOBASlide 68 / 15027 Given circle C below, findD.CA3 501 0 00AnswerEBSlide 69 / 15028 Given circle C below, findDE.C3 50BAnswerA1 0 00
Slide 70 / 15029 Given the figure below, which pairs of angles arecongruent?RASCBUTAnswerDSlide 71 / 15030 FindY.PAnswerXZSlide 72 / 150Answer31 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 73 / 15032 Given circle O, find the value of x.xBA.3 00CDAnswerOSlide 74 / 15033 Given circle O, find the value of x.1 0 00BA.OAnswer3 50CDxSlide 75 / 150Try ThisandIn the circle below,, andFindQ234STAnswerP1
Slide 76 / 150INSCRIBED POLYGONSA polygon is inscribed if all its vertices lie on a EOREM:If a right triangle is inscribed in a circle, then thehypotenuse is a diameter of the circle.Slide 77 / 150A.xiff AC is a diameter of thecircle.LGTHEOREM:A quadrilateral can be inscribed in a circle if and only if itsopposite angles are supplementary.E.NCRN, E, A, and R lie on circle C iffASlide 78 / 150
Slide 79 / 150EXAMPLEFind the value of each variable:K2b2aMAnswerL4b2aJSlide 80 / 15034 The value of x isC6 80A 1500B 980B xD8 20C 1120yD 1800AnswerSlide 81 / 150is a central angle. What is?A 150B 300C 600D 1200AB.DCAnswer35 In the diagram,andA
Slide 82 / 15036 What is the value of x?5B10C13D15(1 2 x 40 0 )(8 x 100 )GFAnswerAESlide 83 / 150Tangents & SecantsReturn to thetable ofcontentsSlide 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).Line 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 / 150Verify A Line is Tangent to a CircleGiven:TIsis a radius of circle Ptangent to circle P?12.35PAnswer}S37Slide 86 / 150Finding the Radius of a CircleIf B is a point of tangency, find the radiusof circle C.5 0 ftr8 0 ft.CAnswerArBTHEOREM:Tangent segments from a common external point are congruent.R.PATIf AR and AT are tangent segments,thenSlide 87 / 150
Slide 88 / 150EXAMPLEGiven: RS is tangent to circle C at S and RT is tangent to circle Cat T. Find x.S28.CTAnswerR3x 4Slide 89 / 15037 AB is a radius of circle A. Is BC tangent to circle A?NoB60C25}.AAnswerYes67Slide 90 / 15038 S is a point of tangency. Find the radius r ofcircle T.31.7B60TC14rD3.5.Sr3 6 cm4 8 cmRAnswerA
Slide 91 / 15039 In circle C, DA is tangent at A and DB is tangent atB. Find x.A25.C3x - 8BAnswerDSlide 92 / 15040 AB, BC, and CA are tangents to circle O. AD 5,AC 8, and BE 4. Find the perimeter of triangleABC.BCAnswerFE.ODASlide 93 / 150Tangents 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 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.M.A.2 1RSlide 95 / 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 tangentstwo secantsBXPA1.Q2CMW3ZYTHEOREM: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 / 150
Slide 97 / 150EXAMPLEFind the value of x.Dx07 601 7 80AnswerCBASlide 98 / 150EXAMPLEFind the value of x.Answer1 3 00x01 5 60Slide 99 / 15041 Find the value of x.C7 80E4 20DFAnswerHx0
Slide 100 / 15042 Find the value of x.(3 x - 02 )(x 60 )Answer3 40Slide 101 / 15043 FindAnswerB6 50ASlide 102 / 1502 6 001Answer44 Find
Slide 103 / 15045 Find the value of x.x.054 50Answer122Slide 104 / 150B2 4 70AnswerTo 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.x0ASlide 105 / 15046 Find the value of x.2 2 00x0AnswerStudents type their answers here
Slide 106 / 15047 Find the value of x.1 0 00x0AnswerStudents type their answers hereSlide 107 / 15048 Find the value of xAnswerStudents type their answers here5 00x0Slide 108 / 15049 Find the value of x.1 2 00(5 x 100 )AnswerStudents type their answers here
Slide 109 / 150(2 x - 300 )Answer50 Find the value of x.x3 00Slide 110 / 150Segments & CirclesReturn to thetable ofcontentsTHEOREM:If two chords intersect inside a circle, then the products of themeasures of the segments of thechords are equal.CAEDBSlide 111 / 150
Slide 112 / 150EXAMPLEAnswerFind the value of x.455xSlide 113 / 150EXAMPLEFind ML & JK.MKLxx 4Answerx 2x 1JSlide 114 / 15051 Find the value of x.16918Answerx
Slide 115 / 15052 Find the value of x.-2B4C5D6x2x 62AnswerAxTHEOREM: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 116 / 150BAEDCSlide 117 / 150EXAMPLE96x5AnswerFind the value of x.
Slide 118 / 15053 Find the value of x.3x 2Answerx 1x- 1Slide 119 / 15054 Find the value of x.4x 4Answer5x- 2THEOREM: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 120 / 150
Slide 121 / 150EXAMPLEFind RS.QRxSAnswer16T8Slide 122 / 15013Answer55 Find the value of x.xSlide 123 / 15056 Find the value of x.12xAnswer24
Slide 124 / 150Equations of aCircleReturn to thetable ofcontentsSlide 125 / 150(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 126 / 150EXAMPLE4AnswerWrite the equation of the circle.
Slide 127 / 150For 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 128 / 150EXAMPLEAnswerWrite the standard equation of a circle withcenter (-2, 3) & radius 3.8.Slide 129 / 150EXAMPLEAnswerThe point (-5, 6) is on a circle with center (-1, 3). Write thestandard equation of the circle.
Slide 130 / 150EXAMPLEThe 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 131 / 15057 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 400Slide 132 / 15058 What is the standard equation of the circle?A (x - 4)2 (y - 3)2 81C (x 4)2 (y 3)2 81D (x 4)2 (y 3)2 9AnswerB (x - 4)2 (y - 3)2 9
Slide 133 / 150A(0,0)B(4,2)C(-4, -2)D(4, -2)Answer59 What is the center of (x - 4)2 (y - 2)2 64?Slide 134 / 150Answer60 What is the radius of (x - 4)2 (y - 2)2 64?Slide 135 / 150A2B4C8D16Answer61 The standard equation of a circle is(x - 2)2 (y 1)2 16.What is the diameter of the circle?
Slide 136 / 150A(-2, -1)B(1, 8)C(3, 4)D(0, 5)Answer62 Which point does not lie on the circle describedby the equation (x 2)2 (y - 4)2 25?Slide 137 / 150Area of a SectorReturn to thetable ofcontentsSlide 138 / 150A sector of a circle is the portion of the circle enclosed by tworadii and the arc that connects them.BMajor SectorMinor SectorAC
Slide 139 / 15063 Which arc borders the minor sector?BAnswerAABCDSlide 140 / 15064 Which arc borders the major sector?WAnswerABXZYSlide 141 / 150Lets think about the formula.The area of a circle is found byWe 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 circleWhen the central angle is in degrees, the fractionof the circle is out of the total 360 degrees.
Slide 142 / 150Finding the Area of a Sector1. Use the formula:Awhen θ is in degreesAnswerBr 3 4 50CSlide 143 / 150Example:Find the Area of the major sector.CAAnswer8 cm6 00TSlide 144 / 15065 Find the area of the minor sector of the circle. Roundyour answer to the nearest hundredth.5 .5 cm 3 00ATAnswerC
Slide 145 / 15066 Find the Area of the major sector for the circle.Round your answer to the nearest thousandth.AnswerC1 2 cmA8 50TSlide 146 / 15067 What is the central angle for the major sector of thecircle?AnswerC1 5 cmA1 2 00GSlide 147 / 15068 Find the area of the major sector. Round to thenearest hundredth.AnswerC1 5 cmA1 2 00G
Slide 148 / 15069 The sum of the major and minor sectors' areas isequal to the total area of the circle.TrueAnswerFalseSlide 149 / 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?Slide 150 / 150Answer71 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?
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
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 / 149
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 .
Coplanar circles that have a common center are called concentric circles. 2 points of intersection 1 point of intersection (tangent circles) no points of intersection concentric circles A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of .
3006 AGMA Toilet Additive 1338 (3006) 19.0% 2914 CERAVON BLUE V10 DC (2914) 0.05% 2922 FORMALDEHYDE REODORANT ALTERNATIVE (2922) 0.6% 3 Water (3) 80.05% Constituent Chemicals 1 Water (3) 80.05% CAS number: 7732-18-5 EC number: 231-791-2 Product number: — EU index number: — Physical hazards Not Classified Health hazards Not Classified Environmental hazards Not Classified 2 Bronopol (INN .
