MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

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STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c yo n l i n e . c o m

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Direction for 1 – 20: State True or False

1. No three points on a circle are collinear.

2. A circle has infinitely many radii.

3. The centre of a circle is a part of the circle.

4. In a circle all the radii have a common end point.

5. An arc is a curved line.

6. The measure of a minor arc is always an acute angle.

7. The region surrounded by an arc and its corresponding chord is called a sector.

8. The region surrounded by an arc and the two radii through the end points of the arc is called a sector.

9. A circle divides the plane on which it lies into two parts.

10. If two arcs of a circle are congruent, their corresponding sectors are also congruent.

11. If two arcs of a circle are equal, then the angles subtended by them at the centre are also equal.

12. There is one and only one circle passing through two distinct points on a plane.

13. There are infinitely many circles passing through distinct points in a plane.

14. If two circles with radii r1 and r2 do not intersect then the distance between their centres is always more than

r1 and r2

15. Any triangle can be inscribed in a circle.

16. The perpendicular bisectors of the chords in a circle are concurrent

17. Two circles are congruent if their diameters are equal.

18. All squares are cyclic.

19. All rectangles are cyclic.

20. All triangles are cyclic.

21. In the figure two circles with centers A and B intersect at P and Q.

Prove that ∠APB = ∠AQB

ASSIGNMENT ON CIRCLES

A

Q

P

B

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

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STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c y o n l i n e . c o m

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22. In the figure, ‘O’ is the centre of the circle. ∠AOB = ∠COD

Prove that arc AC ≅ arc BD.

23. In the figure, AB and CD are two parallel chords. O is the centre of the circle. PQ is the perpendicular

bisector of CD. Prove that PQ bisects AB.

24. Find the length of the chord which is at a distance of 5 cm from the centre of a circle whose radius is

13 cm.

25. PQ and RS are two parallel chords of a circle whose radius is 20 cm. If PQ = 24 cm and RS = 32 cm and

they lie on either side of the centre, find the distance between the chords.

26. In the figure AB and BC are two equal chords and BD is a diameter. If O is the centre of the circle, prove

that BD bisects ∠ABC.

27. Two circles of radii 10 cm and 8 cm intersect at two points. If the length of the common chord is 12 cm,

find the distance between the centers of the circles.

28. In the figure ‘O’ is the centre of the circle and the chords AB and CD intersect at ‘P’. If ∠DPO = ∠APO,

prove that AB = CD

A

C B

D

O

•

A

BO

C

D

A

C

P

O

Q

B

D

•

• O

A

C

P

B D

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

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STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c yo n l i n e . c o m

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29. In the figure if ∠CAB = 60°, find the measure of ∠CPB.

30. In the figure AB is a diameter of the circle. O is the centre of the circle. If ∠BAC = 40°, find ∠ABC and

∠BDC.

31. In the figure AB = CD and O is the centre of the circle. Prove that BE = DE

32. In the figure AB and AC are two equal chords of the circle whose centre is ‘O’. If OD ⊥ AB and OE ⊥

AC, prove that AD = AE.

40° •

A

C

B

D

O

O

A

B

D

C

E •

•

A

D

B

O

E

C

CB

O

A

60°

P

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

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STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c y o n l i n e . c o m

4

33. In the figure AB and CD are two parallel chords in a circle whose centre is ‘O’. If AC is a diameter of

the circle, show that AB = CD.

34. In the figure AB = CD and AB ⊥ CD. If M and N are the mid points of AB and CD. respectively, show

that OMEN is a square.

35. In the figure O is the centre of the circle. If y = 35°, find x and z.

36. In the figure O is the centre of the circle, PQ = QR. Prove that diameter QS bisects ∠PSR and ∠PQR

•

A

D

O

B

C

•

R

Q

O

S

P

• D

B

E

M

N

O

C

•

•

A

•

A

B

y

z C

x

O

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

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STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c yo n l i n e . c o m

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37. In the figure O is the centre of the circle and AB is a diameter. Chord CD is equal to the radius of the

circle. AC produced and BD produced meet at P. Prove that ∠P = 60°.

38. In the figure AB = AC and AD = AE. Prove that BCED is a cyclic quadrilateral.

39. Two diameters of a circle intersect each other at right angles. Prove that the quadrilateral formed by

joining their end points is a square.

40. In the figure ABCD is a cyclic quadrilateral. Find each angle of ABCD.

A

C

O

D

B

P

A

D

B

E

C

3x

5y

y

x

D

C

A

B

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

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STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c y o n l i n e . c o m

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41. In the figure ABC is an isosceles triangle with AC = BC. If ∠ABC = 38°, find ∠ADB and ∠AEB.

42. In the figure O is the centre of the circle. Find the value of x

43. In the figure ‘O’ is the centre of the circle. If ∠ADC = 140°. Find ∠CAB.

44. In the figure chords AC and BD intersect at E. If ∠AEB = 110° and ∠CBD = 25°, find ∠ADB and

∠APB

A

C

E

D

B38°

A

B

C D

x O

130°

•

D

A O

C

B

140°

A

D

P

B

E

C

110° 25°

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

_________________________________________________________________________________

STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c yo n l i n e . c o m

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45. ABC and ADC are two right triangles with common hypotenuse AC. Show that ∠CAD = ∠CBD

46. AC and BD are two chords of a circle which bisect each other. Prove that AC and BD are diameters.

47. Prove that the line segment joining the centres of two intersecting circles bisects the common chord.

48. Prove that the circle drawn with any side of rhombus as diameter will pass through the point of

intersection of its diagonals.

49. In the figure ABCD is a cyclic quadrilateral with AD ⎟⎟ BC. Prove that AB = CD.

50. Two circles intersect at two points A and B. If AD and AC are the diameters of the circles, prove that B

lies on the line segment DC.

A

B C

D

A

D B C

• • O P

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

_________________________________________________________________________________

STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c y o n l i n e . c o m

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ANSWER KEY

CIRCLES

1. True 2. True 3. False 4. True

5. True 6. False 7. False 8. True

9. False 10. True 11. True 12. False

13. False 14. False 15. True 16. True

17. True 18. True 19. True 20. True

24. 24 cm 25. 28 cm 27. 13.3 cm 29. 120°

30. 58°, 40° 35. 55°, 110° 41. 104°, 76° 42. 25°

43. 50o 44. 85°, 95°