1. 1
4 Circle

2. Std : IX Subject : Geometry
4 Circle
Std : IX Subject : Geometry
Introduction : Respected sir, in previous standards we have already taught the techniques of drawing a circle of given radius , the distance between the centre of the circle and the chord , property of perpendicular drawn from the centre of the circle to its chord., the property of distances between the centre and congruent chords of that circle.

3. Comparison between previous units and upgraded units of circle
Now in upgraded syllabus for IX Std we study the definition of circle and some concepts related to circle , its properties with proof using the concept of constructivist approach.
Comparison between previous units and upgraded units of circle
Units- arc of a circle, types of arc, central angle, angular measure of arc, angle subtended by arc in previous syllabus are deleted and included in the upgraded syllabus of X th std.
But the theorem , there is one and only one circle passing through the three non collinear points is newly added.

4. Subunits : 4.1 Circle , related terms
4.2 : points in a plane of circle
4.3 : Circle passing through given points
4.4 : Circles in a plane
Concentric circles b) Congruent circles
c) Intersecting circles
4.5 Circle and line in a plane.
4.6 : Angle subtended by chord ( Theorem and converse)
Teacher’s and student’s activity for developing the concept.
Teacher : 1)Comes in the classroom carrying different shapes cards. On each card point M is shown.
2) Make groups of 15 students each.

5. Here distances of points P , Q , R , S from point M are not equal.
3) Ask the students to find the distance of points P,Q,R , and S etc. from point M.
Group-1 In fig MP = 4 cm
MQ = 3 cm ,
MR = 4.5 cm
MS = 2.5 cm
Here distances of points P , Q , R , S from point M are not equal. Q P R M . S

6. Here distances of points P , Q , R , S,G,N from point M are not equal.
Group-2 In fig. MR = 4 cm
MQ = 4 cm ,
MP = 4 cm
MS = 4 cm
but MG=MN= 3cm
Here distances of points P , Q , R , S,G,N from point M are not equal. S P R G N Q M .

7. Here distances of points P , Q , R , S,T,U from point M are equal.
Group-3
In fig. MP = MQ
MR = MS = MT
MU = 3 cm
Here distances of points P , Q , R , S,T,U from point M are equal. U P S . T Q R M

8. I summarized the outcome of their achievement and tell the students in classroom ,’yes ’ in a circle ( fig for group 3) the points on periphery are equidistant from the fix point M .Here M is the centre of the circle and periphery is the circumference of the circle.
Defination of circle : A circle is a locus of points in a plane which are a constant distance from a fix point in that plane.
Here fix point is called the centre of the circle.

9. and constant distance is called the radius of the circle
and constant distance is called the radius of the circle. The radius is shown by the line segment from the centre to the circumference. A . B M r
The circle consist of all those points whose distance from the centre of the circle equals the radius. Diameter of the circle d = 2r

10. Do You know
The circle is special case of ellipse in which two foci are coincident. f . f1 f2 .

11. 4.1 Here Point O is the centre of the circle. B A O C l m D
Now we can recall previous knowledge of students which are obtained from above activities.
4.1 Here Point O is the centre of the circle.
Seg OC seg AD line m segAB line l

12. Points in plane of the circle . .T A O D .G .F B .Q .P .M .L
Points in plane of the circle
Circle is a simple closed figure which divides the plane into three disjoint parts.
Sr.no Position Name of the points
1 In the interior ………………..
2 In the exterior ……………………
3 On the circle ……………………

13. Point O is the centre and r is the radius of the circle If
OA = r then A is on the circle
OA > r then A is in the exterior of the circle
OA< r then A is in the interior of the circle . .T A O D .G .F B .Q .P .M .L

14. Circular region :
The union of a circle and its interior is called the circular region.

15. Circles passing through given points .
1)Take a point A in a plane. Draw circles passing through point A. How many circles can be drawn passing through a given point? . A

16. In second activity, Take two points A and B in the plane.
Draw circles passing
through these two points.
How many circles can be drawn passing through the given two
points?
Where will the centre or centers lies ?
What will be the radii or radius

17. Draw seg AB, draw perpendicular bisector of seg AB.
Take any point on the bisector as a centre and draw circles
passing through A and B.
How many circles will pass through the points A and B? A B

18. Take any three collinear points A, B and C in a plane.
Draw circle or circles passing through all these three points.
How many circles will pass through these three collinear points?

19. Take any three non-collinear points P,Q,R in a plane.
How many circles will pass through all these points?
Where will be the centre of the circle?
Join three non collinear points. we get DPQR. C R Q P

20. Draw perpendicular bisectors of the sides of a triangle.
perpendicular bisectors intersect in the point C. where PC=CR and QC= CR why?
Take C as a centre, draw the circle passing through points p, Q, R?
Three non collinear points determine a unique circle. C R Q P

21. Theorem 4.1: There is one and only one circle passing through the three non-collinear points'B C m l
To prove : There is one and only one circle passing through the
points A, B, C.
Construction : Join AB and BC
Draw perpendicular bisectors / and m of seg AB and segBC
Respectively. Line / and line m intersect in point O.

22. Proof: A, B, C are non collinear points
.'. seg AB and seg BC are not on the same line.
.'. line / is not parallel to line m.
.'. line l and line m intersect in one and only one point O.
Every point on the line / is equidistant from end points of the seg ment AB
.'. OA = OB (i)
Similarly, O is a point on line m
.'. OB = OC. ….....(ii)
.'. OA = OB= OC From (i) and (ii)
Let OA = OB= OC = r (i)

23. .'. Points A, B, C are such points on the circle whose centre is O and radius is r.
.'. There exists a circle passing through the three non collinear points.
Now we will prove its uniqueness.
Let O' be the centre and r' be the radius of the circle which passes through the
points A, B and C.
To prove O = O' and r = r'
.'. O'A = O'B = O'C = r‘…(iv) consideration
From (iii) and (iv) O and O' are the points which are on the seg AB and seg BC.

24. .'. O and O' are intersecting points of line / and line m.
This is the contradiction.
.'. Point O and O' are the same points similarly
r‘= O'A = OA= r
.'. Circle passing through the points A, B, C is unique.

25. Circles in a plane: Given two circles in a Plane,
there are following possibilities
Concentric circles:
Circles having same centre
concentric circles and different radii are called Concentric circles. O B A

26. 2) Congruent circles:
Circles having equal radii are called congruent circles.
Here PA and QB are circles with centers P and Q
are congruent circles
As PA = QB P A Q B

27. Intersecting circles:
l) coplanar circles having two points in common are called intersecting circles.
Here circles with centers P and Q are intersecting each other in two points namely A and B. B A .Q .P

28. ii) Coplanar circles intersecting in one
point are called touching circles.
In fig 4.15 and 4.16 circles with centres P and Q are touching in point A. A .P .Q A .Q .P

29. Circle and a line in the Plane
Write relation between a circle and a line given in the In fig (a) ,circle and
line l are disjoint sets.
In fig (b) line / is interesting the circle with
centre O in only one Point,
In fig (c) line l is intersecting the circle with centre O in two points A and B.
In this case line I is called a "secant" .O l .P B A .Q P l

30. Secant: The line in the plane of the circle which intersects the circle in two distinct points is called a secant.
Note:
ln fig (b) seg AB is a chord. Every secant of a circle contains chord.
:Diameter : Chord passing through the centre of the circle is called a diameter.
Diameter is the largest chord of the circle.
Every diameter is a chord, But Every chord
is not a diameter.

31. Angle Subtended by the chord
Angle subtended by the chord at the centre.
ÐAOB is the angle subtended by chord AB at the centre O.
Angle subtended by the chord at any point on the circle. ÐAPB is the angle subtended chord AB at point P. O A B P A B .O

32. Angle subtended by the chord any point inside the circle.
ÐAPB is the angle subtended chord AB at point P in the interior of the circle.
(iii) Angle subtended by the chord any point outside the circle.
ÐAPB is the angle subtended chord AB at point P in the exterior of the circle. A B P .O P A B .O

33. if AB is a diameter then guess the measure of an angle subtended
Activity
if AB is a diameter then guess the measure of an angle subtended
by the diameter at any point on the circle. Take different positions of
point P and draw a conclusion.
mÐAPB = 900 P A B O P1 P2 P3 P4 .

34. Theorem 4.2: The perpendicular from the centre of a circle to a chord bisects the chord.
Converse 4.3:The segment joining the mid point of a chord and the centre of a circle is perpendicular to the chord. C M D P

35. Theorem 4.4: The perpendicular from the centre of a circle to a chord bisects the chord.
Theorem 4.5 :The segment joining the mid point of a chord and the centre of a circle is perpendicular to the chord. C M D P

36. Theorem 4.4 : Congruent chords of congruent circle are equidistant from the centre.P M A B Q N C D
Theorem 4.5 Converse : If the angles subtended by chords at the centre of the circle are congruent , then the corresponding chords are equal.

37. Theorem : Chords which are equidistant from the centre of the congruent circle are congruent.P M A B Q N C D

38. Theorem : If the angles subtended by chords at the centre of a circle are congruent then the corresponding chords are congruent. O Q R S P

39. Questions carrying 1,2,3,4,5 marks
Q.1: (1) Observe the given figure . Which points are in the interiOr and exterior (1)
2) Two circles intersect each other in two distinct points .How many common chords are there ? (1)
Q.2 :1) Radius of the circle with centre O is 7 cm , if OP= 4 cm and OQ = 9 cm then write which point is in the interior and exterior with reason. (2) . .T O .M .N .S .P .N

40. 2) Points A,B,C,D are on the circle with centre O, write the names of any three radii and any three chords of a circle. (2)
3) The radius of the circle is 8Ö2 cm .Find the length of the largest chord of the circle. (2)
Q.3 : The radius of the circle with centre P is 25 cm and the length of the chord is 48 cm. Find the distance of the chord from the centreP of the circle. (3)
2) In fig AP = 34 , AM = 30 if segAM ^segPQ .Find the length of the chord PQ (3) Q P M A

41. 3) If the diameter of the circle bisect each of the two chords of a circle then prove that the chords are parallel. (3)
Q.4 : 1) Prove that the congruent chords of one circle are equidistance from the centre of that circle. (4)
2) Prove that the perpendicular drawn from the centre of the chord bisect the chord. (4)

42. 3) In given figure , chord AB // chord CD of a circle with centre O and radius is 5 cm such that AB = 6 cm and CD = 8 cm , if OP ^ AB , OQ ^ CD .Determine PQ. (4)
Q.5 :1) A boat in a circular lake lies at the centre about 10 m away from the bridge lying in 40 m distance across the circular lake. How much distance will boat have to travel to reach the extreme point of the left of the bridge.(5) B A D C O Q P

43. 2) For Lazim Kawayat for girls if std IX th are arranged in two circular rings as shown in fig
If PA = 15 , PR = 25 , QR = 10 then find RS , PQ and QT. (5)
3) A chord AB of a circle of distance 8 cm less than the radius from the centre .If the radius of a circle is less than chord by 11 cm ,then find the length of the chord , radius and distance of a chord from the centre. (5) P Q R S T A

44. Unit Test Examination Class : IX th Time : 40 min Marks =20
Q.1: If the radius of a circle is 4 cm , find its diameter
Q.2: Observe the following figure and state which points are in the interior , exterior and on the circle . .T P .N .S .K .L M.

45. Q. 3: If Point P is the centre of the circle with radius 6
Q.3: If Point P is the centre of the circle with radius 6.7 cm , d(P,Q)= 7.6 cm , d(P,R) = 5.7 cm. Find the position of the points Q and R with respect to the circle
Q.4: Prove that the perpendicular drawn from the centre of a circle to its chord bisect the chord

46. Show that ABC is isosceles.
Q.5 : In given figure
OA = 5 cm , OM = 3 cm ,
find the length of the
chord AB
Q.6: In figure, seg AB is a
chord of a circle with
centre O. Diameter CD
is perpendicular to AB
intersecting it at E.
Show that ABC is isosceles. O A M B 3 5 D A B C E

47. Circle Lesson