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C IRCLES 169,10 2 Circles and Its Related Terms A Review. Take a compass and fix a pencil in it Put its pointed. leg on a point on a sheet of a paper Open the other. leg to some distance Keeping the pointed leg on the. same point rotate the other leg through one revolution. What is the closed figure traced by the pencil on,paper As you know it is a circle see Fig 10 2 How. did you get a circle You kept one point fixed A in. Fig 10 2 and drew all the points that were at a fixed. distance from A This gives us the following definition. The collection of all the points in a plane, which are at a fixed distance from a fixed point in. the plane is called a circle, The fixed point is called the centre of the circle. and the fixed distance is called the radius of the. circle In Fig 10 3 O is the centre and the length OP. is the radius of the circle,Remark Note that the line segment joining the.
centre and any point on the circle is also called a. radius of the circle That is radius is used in two Fig 10 3. senses in the sense of a line segment and also in the. sense of its length,You are already familiar with some of the. following concepts from Class VI We are just,recalling them. A circle divides the plane on which it lies into,three parts They are i inside the circle which is. also called the interior of the circle ii the circle. and iii outside the circle which is also called the Fig 10 4. exterior of the circle see Fig 10 4 The circle and. its interior make up the circular region, If you take two points P and Q on a circle then the line segment PQ is called a. chord of the circle see Fig 10 5 The chord which passes through the centre of the. circle is called a diameter of the circle As in the case of radius the word diameter. is also used in two senses that is as a line segment and also as its length Do you find. any other chord of the circle longer than a diameter No you see that a diameter is. the longest chord and all diameters have the same length which is equal to two. File Name C Computer Station Maths IX Chapter Chap 10 Chap 10 03 01 2006 PM65. 170 MATHEMATICS,times the radius In Fig 10 5 AOB is a diameter of.
the circle How many diameters does a circle have,Draw a circle and see how many diameters you can. A piece of a circle between two points is called, an arc Look at the pieces of the circle between two. points P and Q in Fig 10 6 You find that there are Fig 10 5. two pieces one longer and the other smaller,see Fig 10 7 The longer one is called the major. arc PQ and the shorter one is called the minor arc. PQ The minor arc PQ is also denoted by p,the major arc PQ by qPRQ where R is some point on. the arc between P and Q Unless otherwise stated,arc PQ or p.
PQ stands for minor arc PQ When P and, Q are ends of a diameter then both arcs are equal Fig 10 6. and each is called a semicircle,The length of the complete circle is called its. circumference The region between a chord and, either of its arcs is called a segment of the circular. region or simply a segment of the circle You will find. that there are two types of segments also which are. the major segment and the minor segment,see Fig 10 8 The region between an arc and the. two radii joining the centre to the end points of the. arc is called a sector Like segments you find that Fig 10 7. the minor arc corresponds to the minor sector and the major arc corresponds to the. major sector In Fig 10 9 the region OPQ is the minor sector and remaining part of. the circular region is the major sector When two arcs are equal that is each is a. semicircle then both segments and both sectors become the same and each is known. as a semicircular region,Fig 10 8 Fig 10 9, File Name C Computer Station Maths IX Chapter Chap 10 Chap 10 03 01 2006 PM65.
C IRCLES 171,EXERCISE 10 1,1 Fill in the blanks, i The centre of a circle lies in of the circle exterior interior. ii A point whose distance from the centre of a circle is greater than its radius lies in. of the circle exterior interior, iii The longest chord of a circle is a of the circle. iv An arc is a when its ends are the ends of a diameter. v Segment of a circle is the region between an arc and of the circle. vi A circle divides the plane on which it lies in parts. 2 Write True or False Give reasons for your answers. i Line segment joining the centre to any point on the circle is a radius of the circle. ii A circle has only finite number of equal chords. iii If a circle is divided into three equal arcs each is a major arc. iv A chord of a circle which is twice as long as its radius is a diameter of the circle. v Sector is the region between the chord and its corresponding arc. vi A circle is a plane figure,10 3 Angle Subtended by a Chord at a Point. Take a line segment PQ and a point R not on the line containing PQ Join PR and QR. see Fig 10 10 Then PRQ is called the angle subtended by the line segment PQ. at the point R What are angles POQ PRQ and PSQ called in Fig 10 11 POQ is. the angle subtended by the chord PQ at the centre O PRQ and PSQ are. respectively the angles subtended by PQ at points R and S on the major and minor. Fig 10 10 Fig 10 11, Let us examine the relationship between the size of the chord and the angle. subtended by it at the centre You may see by drawing different chords of a circle and. File Name C Computer Station Maths IX Chapter Chap 10 Chap 10 03 01 2006 PM65. 172 MATHEMATICS, angles subtended by them at the centre that the longer.
is the chord the bigger will be the angle subtended. by it at the centre What will happen if you take two. equal chords of a circle Will the angles subtended at. the centre be the same or not,Draw two or more equal chords of a circle and. measure the angles subtended by them at the centre. see Fig 10 12 You will find that the angles subtended. by them at the centre are equal Let us give a proof. of this fact Fig 10 12, Theorem 10 1 Equal chords of a circle subtend equal angles at the centre. Proof You are given two equal chords AB and CD,of a circle with centre O see Fig 10 13 You want. to prove that AOB COD,In triangles AOB and COD,OA OC Radii of a circle. OB O D Radii of a circle,AB CD Given,Therefore AOB COD SSS rule.
This gives AOB COD Fig 10 13,Corresponding parts of congruent triangles. Remark For convenience the abbreviation CPCT will be used in place of. Corresponding parts of congruent triangles because we use this very frequently as. you will see, Now if two chords of a circle subtend equal angles at the centre what can you. say about the chords Are they equal or not Let us examine this by the following. Take a tracing paper and trace a circle on it Cut, it along the circle to get a disc At its centre O draw. an angle AOB where A B are points on the circle,Make another angle POQ at the centre equal to. AOB Cut the disc along AB and PQ,see Fig 10 14 You will get two segments ACB.
and PRQ of the circle If you put one on the other,what do you observe They cover each other i e. they are congruent So AB PQ Fig 10 14, File Name C Computer Station Maths IX Chapter Chap 10 Chap 10 03 01 2006 PM65. C IRCLES 173, Though you have seen it for this particular case try it out for other equal angles. too The chords will all turn out to be equal because of the following theorem. Theorem 10 2 If the angles subtended by the chords of a circle at the centre. are equal then the chords are equal, The above theorem is the converse of the Theorem 10 1 Note that in Fig 10 13. if you take AOB COD then,AOB COD Why,Can you now see that AB CD.
EXERCISE 10 2, 1 Recall that two circles are congruent if they have the same radii Prove that equal. chords of congruent circles subtend equal angles at their centres. 2 Prove that if chords of congruent circles subtend equal angles at their centres then. the chords are equal,10 4 Perpendicular from the Centre to a Chord. Activity Draw a circle on a tracing paper Let O, be its centre Draw a chord AB Fold the paper along. a line through O so that a portion of the chord falls on. the other Let the crease cut AB at the point M Then. OMA OMB 90 or OM is perpendicular to,AB Does the point B coincide with A see Fig 10 15. Yes it will So MA MB Fig 10 15, Give a proof yourself by joining OA and OB and proving the right triangles OMA.
and OMB to be congruent This example is a particular instance of the following. Theorem 10 3 The perpendicular from the centre of a circle to a chord bisects. What is the converse of this theorem To write this first let us be clear what is. assumed in Theorem 10 3 and what is proved Given that the perpendicular from the. centre of a circle to a chord is drawn and to prove that it bisects the chord Thus in the. converse what the hypothesis is if a line from the centre bisects a chord of a. circle and what is to be proved is the line is perpendicular to the chord So the. converse is, File Name C Computer Station Maths IX Chapter Chap 10 Chap 10 03 01 2006 PM65. 174 MATHEMATICS, Theorem 10 4 The line drawn through the centre of a circle to bisect a chord is. perpendicular to the chord, Is this true Try it for few cases and see You will. see that it is true for these cases See if it is true in. general by doing the following exercise We will write. the stages and you give the reasons,Let AB be a chord of a circle with centre O and. O is joined to the mid point M of AB You have to,prove that OM AB Join OA and OB.
see Fig 10 16 In triangles OAM and OBM,OA OB Why Fig 10 16. OM OM Common,Therefore OAM OBM How,This gives OMA OMB 90. 10 5 Circle through Three Points, You have learnt in Chapter 6 that two points are sufficient to determine a line That is. there is one and only one line passing through two points A natural question arises. How many points are sufficient to determine a circle. Take a point P How many circles can be drawn through this point You see that. there may be as many circles as you like passing through this point see Fig 10 17 i. Now take two points P and Q You again see that there may be an infinite number of. circles passing through P and Q see Fig 10 17 ii What will happen when you take. three points A B and C Can you draw a circle passing through three collinear points. File Name C Computer Station Maths IX Chapter Chap 10 Chap 10 03 01 2006 PM65. C IRCLES 175, No If the points lie on a line then the third point will. lie inside or outside the circle passing through two. points see Fig 10 18, So let us take three points A B and C which are not on the same line or in other.
words they are not collinear see Fig 10 19 i Draw perpendicular bisectors of AB. and BC say PQ and RS respectively Let these perpendicular bisectors intersect at. one point O Note that PQ and RS will intersect because they are not parallel see. Fig 10 19 ii, Now O lies on the perpendicular bisector PQ of AB you have OA OB as every. point on the perpendicular bisector of a line segment is equidistant from its end points. proved in Chapter 7, Similarly as O lies on the perpendicular bisector RS of BC you get. So OA OB OC which means that the points A B and C are at equal distances. from the point O So if you draw a circle with centre O and radius OA it will also pass. through B and C This shows that there is a circle passing through the three points A. B and C You know that two lines perpendicular bisectors can intersect at only one. point so you can draw only one circle with radius OA In other words there is a. unique circle passing through A B and C You have now proved the following theorem. Theorem 10 5 There is one and only one circle passing through three given. non collinear points, File Name C Computer Station Maths IX Chapter Chap 10 Chap 10 03 01 2006 PM65. 176 MATHEMATICS, Remark If ABC is a triangle then by Theorem 10 5 there is a unique circle passing. through the three vertices A B and C of the triangle This circle is called the. circumcircle of the ABC Its centre and radius are called respectively the. circumcentre and the circumradius of the triangle, Example 1 Given an arc of a circle complete the circle.
Solution Let arc PQ of a circle be given We have, to complete the circle which means that we have to. find its centre and radius Take a point R on the arc. Join PR and RQ Use the construction that has been, used in proving Theorem 10 5 to find the centre and. Taking the centre and the radius so obtained we,can complete the circle see Fig 10 20. EXERCISE 10 3, 1 Draw different pairs of circles How many points does each pair have in common. What is the maximum number of common points, 2 Suppose you are given a circle Give a construction to find its centre.
3 If two circles intersect at two points prove that their centres lie on the perpendicular. bisector of the common chord, 10 6 Equal Chords and Their Distances from the Centre. Let AB be a line and P be a point Since there are, infinite numbers of points on a line if you join these. points to P you will get infinitely many line segments. PL1 PL2 PM PL 3 PL4 etc Which of these is the,distance of AB from P You may think a while and. get the answer Out of these line segments the,perpendicular from P to AB namely PM in Fig 10 21. will be the least In Mathematics we define this least. length PM to be the distance of AB from P So you,may s.
CIRCLES 10 1 Introduction You may have come across many objects in daily life which are round in shape such as wheels of a vehicle bangles dials of many clocks coins of denominations 50 p Re 1 and Rs 5 key rings buttons of shirts etc see Fig 10 1 In a clock you might