Line OB and OA are theradii of the circle DAB and CD are chordsof the circle CF is also the chord ofthe cirle known as C O FDIAMETERDiameter is the longest - A B---------------- of thecircle
Area in green part isknown as major sectorArea in minor part isknown as -----------------And the arc comprisedin these sectors arerespectively known asMajor arcMinor arc.
Angle ABC is subtendedangle in circle withcentre oAngle DOE is the centralangle as it is makingangle at the centre. Angles made in circle : the angles lying anywhere ON the the circle made by chords is known as SUBTENDED angle ( line AC is the chord)
A segment is any regionin a circle separated by achordPortion in green regionis known as the MajorsegmentPortion in purple color isknown as minorsegmentWhat is the segmentseparated by a diameter Major segment , minor segment andknown as?? Semicircles
Quick recap of Aall the termsFrom the figure asidename the following :1. Points in the interior ofthe circle2. Diameter of the circle O B3.Radius of the circle4.Subtended angle in thecircle5.Central angle in thecircle C6.Major sector D7.Minor sector8.Semicricle
Equal chords of a circle subtend equal angles at the centre
Given: Chord AB = chord DCTo Prove: A Dangle AOB= angle DOCProof:In Triangle ABC and triangleDOC OOAB=DC givenAO=OC radii of same circleBO=OD radii of same circle C BTriangle AOB= Triangle DOCangle AOB= angle DOC(C.P.C.T) Equal chords of a circle subtendHence proved……. equal angles at the centre
Given :Angle AOB= angle CODTo prove: A Bchord AB= Chord CD `Proof:In triangle AOB and triangleCOD CAngle AOB= angle COD (given ) OAO=OC radii of same circleBO=OD radii of same circleTriangle AOB= TriangleDOC Dchord AB= Chord CD If the angles subtended by the chords of a circle at the centre are congruent , then the chords are congruent.
Given :OD perpendicular ABTo prove:AD=DBProof:In triangle AOD andtriangle DOB OOA=OB radiusOD=OD common sideAngle ODA=angle ODB A D B(90 degrees.)Triangle AOD=ODB(R-H-S test) The perpendicular from the centre of the circle bisects the chord.AD=DB ( C.P.C.T)
Given : AD=DBTo prove: ODperpendicular ABProof:In triangle AOD andtriangle DOB OOA=OB radiusOD=OD common sideAD=DB giventriangle AOD = triangle ADOB S-S-S test D BAngle ODB=OAD(C.P.C.T)Angle ODB+angleOAD=180 linear pair The line drawn through the centreAngle ODB= ½ angleADB of a circle to bisect the chord isAngle ODB=90 perpendicular to the chord
Circle through 1,2,3, points On a sheet of paper try drawing circle through one point Two points Three points What do you see?
Answers Many circles can be drawn from one point Many circles can be drawn from two points But one and only one circle can be drawn from three points.
Try naming them andproving it. OOD is perpendicular tothe lineOthers are allhypotenuseIn a right angle trianglehypotenuse is thelongest side… DSo…………………………………………. The length of the perpendicular from a point to a line is the (shortest) distance of the line from the centre
Given: AB=CDTo prove: OF=OE CDraw OF perpendicularto OE A O OOO E F D B Equal chords of a circle (or congruent circles) are equidistant from the centre
Pick statements in proper order to prove the theorem and match the reasons Statements Reasons AF=FB Radii of same circle AF=1/2AB C.P.C.T CE=ED Given CE=1/2CD Radii of same circle CE=AF S-S-S test Chord AF=chord CE S-A-S test OA =OC Each 90 degrees OB=OD In triangles AOF and OCE Triangles congruent by Angle F= Angle E OF=OE
Chords Equidistant from the centre of a circle are equal in length (converse of the earlier theorem) Try proving this………………….. Have fun
Concentric circles : Circle with same centre are known as concentric circ ooo
MThe angels subtended byan arc at the centre isdouble the anglesubtended by it at anypoint on the remainingpart of the circle oAngle . AMB is half ofangle AOBAngle AOB= angle of arcACB A BAngle AMB= ½ of arcAMB C Angles Subtended by an Arc of a chord.
Angles ADB CACBAEB EAll lie in arc AMB DHence all are equal to ½arc AMBSo angle A B ADB =ACB=AEB=1/2 Marc AMB Angles in the same segment of a circle are equal
CyclicQuadrilateralsA Quadrilateral whose4 corners are on sides ofthe circle is known ascyclic Quadrilateral
Properties of 1. the sum of either pair ofCyclic opposite angles of a cyclicQuadrilateral quadrilateral is 180 degrees If the sum of opposite angles of a quadrilateral is 180 degrees its cyclic quadrilateral.