CIRCLE. Circle is a collection of all points in a plane which are equidistant from a fixed point.

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2 CIRCLE Circle is a collection of all points in a plane which are equidistant from a fixed point. The fixed point is called as the centre and the constant distance is called as the radius.

3 Parts of a Circle F centre

4 PARTS OF A CIRCLE tangent diameter chord secant radius

5 1. The line AB and the circle have no common point. In this case, AB is called a non-intersecting line with respect to the circle. 2. There are two common points A and B that the line and the circle have in this case, we called the line AB a secant of the circle. 3. There is a only one point P which is a common to the line AB and the circle. In this case, the line is called a tangent to the circle.

6 Secant A A line that intersects the circle at exactly two points. O C Example: AB D B

7 SECANT If a line intersects the circle in two distinct points A and B, then it is called Secant. In this case Perpendicular distance of this line from Centre of the circle is less than the radius of the circle. such that OP < r.

8 Tangent B A C A line that intersects a circle at exactly one point. Example: AB

9 TANGENT TO A CIRCLE If a line intersects the circle at one common point P, then it is called Tangent to the Circle. In this case Perpendicular distance of this line from Centre of the circle is equal to the radius of the circle such that OP = r. O r P

10 NUMBER OF TANGENTS TO A CIRCLE FROM A (i) When P lies inside the circle : POINT No tangent can be drawn from a point P lying inside the circle. Every line drawn through P is a Secant line intersecting the circle at two points.

11 ii) When P lies on the circle : Exactly one tangent can be drawn to the circle touching it at point P. Rest all the lines passing through P will intersect the circle at two points.

12 iii) When P lies outside the circle : Exactly two tangents can be drawn from a point P to the circle touching it at two points Q and R. P R Q

13 PROPERTIES OF TANGENT TO A CIRCLE The tangent at any point of a circle is perpendicular to the radius through the point of contact. The length of tangents from an external point to a circle are equal.

14 THEOREM 01 STATEMENT: The tangent at any point of a circle is perpendicular to the radius through the point of contact. P Q O A

15 Given : AP is a tangent drawn from point A to a circle with centre O. P To Prove : APO = 90⁰ A Q O Construction : Take another point Q on the tangent. Join OP and OQ.

16 Proof: Obviously, point Q will be outside the circle. Therefore, OP< OQ OP will be the shortest distance from the centre to the tangent AP. OP will be perpendicular to AP. Therefore, APO = 90⁰. ( Hence Proved )

17 THEOREM 02 STATEMENT: The length of tangents drawn from an external point to a circle are equal.

18 GIVEN : A CIRCLE WITH CENTRE O. AP AND AQ ARE TWO TANGENTS DRAWN FROM EXTERNAL POINT A. TO PROVE : AP = AQ

19 CONSTRUCTION: JOIN OP, OQ AND OA FIGURE: P A O Q

20 PROOF: IN OAP AND OAQ OP = OQ----[ RADII OF SAME CIRCLE ] OA = OA ----[ COMMON ] P O A Q OPA = OQA---[ RADIUS THROUGH POINT OF CONTACT IS PERPENDICULAR TO TANGENT ] HENCE OAP OAQ----[ SSS ] AP = AQ [ CPCT ]

21 APPLICATION OF THEOREMS Show that the tangent lines at the end points of diameter of a circle are parallel. R B S O P A Q

22 Let AB be a diameter of a given circle, and let PQ and RS be the tangent lines drawn to the circle at points A and B respectively. Since tangent at a point to the circle is perpendicular to the radius through the point. Therefore, AB is perpendicular to PQ and AB is perpendicular to RS. PAB = 90⁰ and ABS = 90⁰ PAB = ABS PQ RS ( Since PAB and ABS are alternate angles) ( Hence Proved )

23 RECAPITULATION LEVEL I 1) WHAT IS A TANGENT? 2)AT HOW MANY POINTS DOES A SECANT INTERSECT A CIRCLE? 3) CAN YOU DRAW A TANGENT FROM A POINT OUTSIDE A CIRCLE? IF SO, HOW MANY TANGENTS CAN YOU DRAW?

24 LEVEL II 1) WHAT IS THE RELATION BETWEEN THE LENGTHS OF TWO TANGENTS DRAWN FROM EXTERNAL POINTS? 2) HOW MANY PARALLEL TANGENTS CAN YOU DRAW IN A CIRCLE? 3) FIND THE LENGTH OF A TANGENT FROM A POINT WHICH IS AT A DISTANCE OF 5 CM FROM THE CENTRE OF A CIRCLE OF RADIUS 3 CM. 4) FIND THE RADIUS OF A CIRCLE IN WHICH A 12 CM LONG TANGENT IS DRAWN FROM A POINT AT A DISTANCE OF 13 CM FROM CENTRE.

25 LEVEL III 1) IN TWO CONCENTRIC CIRCLES, PROVE THAT A CHORD OF LARGER CIRCLE WHICH IS A TANGENT TO A SMALLER CIRCLE IS BISECTED AT THE POINT OF CONTACT. 2) TWO CONCENTRIC CIRCLES ARE OF RADII 5 CM AND 3 CM. FIND OUT THE LENGTH OF THE CHORD OF LARGER CIRCLE WHICH TOUCHES THE SMALLER CIRCLE. 3)IF ALL SIDES OF A PARALLELOGRAM TOUCH A CIRCLE, SHOW THAT THE PARALLELOGRAM IS A RHOMBUS.

26 4) ABC IS A RIGHT TRIANGLE RIGHT ANGLED AT B SUCH THAT BC = 6 CM AND AB = 8 CM, FIND THE RADIUS OF THE INCIRCLE. 5)A circle touches all the four sides of a quadrilateral ABCD. Prove that: AB+CD=BC+DA. A P B S Q D R C

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