assignments in mathematics class ix (term 2) 10. circles · l the collection of all the points in a...

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GOYAL B

ROTHERS PRAKASHAN

10. CIRCLES

IMPORTANT TERMS, DEFINITIONS AND RESULTS

l The collection of all the points in a plane, whichare at a fixed distance from a fixed point in the plane,is called a circle.

l The fixed point is called the centre of the circleand the fixed distance is called the radius of thecircle.

In the given figure, O is the centre and the lengthOP is the radius of the circle.

l A circle divides the plane on which it lies into threeparts. They are : (i) inside the circle, which is alsocalled the interior of the circle; (ii) the circle and(iii) outside the circle, which is also called theexterior of the circle. The circle and its interiormake up the circular region.

l A chord of a circle is a line segment joining anytwo points on the circle. In the given figure PQ, RSand AOB are the chords of a circle.

l A diameter is a chord of a circle passing throughthe centre of the circle. In the given figure, AOB isthe diameter of the circle. A diameter is the longestchord of a circle.

Diameter = 2 × radiusl 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 the given figure. You find thatthere are two pieces, one longer and the othersmaller. The longer one is called the major arc PQand the shorter one is called the minor arc PQ.

l The length of the complete circle is called itscircumference. The region between a chord andeither of its arcs is called a segment of the circularregion or simply a segment of the circle. You willfind that there are two types of segments also, whichare the major segment and the minor segment.

l The region between an arc and the two radii, joiningthe centre to the end points of the arc is called asector. Like segments, you find that the minor arccorresponds to the minor sector and the major arccorresponds to the major sector.

l Equal chords of a circle subtend equal angles at thecentre.

l If the angles subtended by the chords of a circle atthe centre are equal, then the chords are equal.

l The perpendicular from the centre of a circle to achord bisects the chord.

l The line drawn through the centre of a circle to bisecta chord is perpendicular to the chord.

l There is one and only one circle passing throughthree given non-collinear points.

Assignments in Mathematics Class IX (Term 2)

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GOYAL B

ROTHERS PRAKASHAN

SUMMATIVE ASSESSMENT

MULTIPLE CHOICE QUESTIONS [1 Mark]

A. Important Questions1. In the figure, O is the centre of the circle with AB

as diameter. If ∠AOC = 40°, the value of x isequal to :

(a) 50° (b) 60° (c) 70° (d) 80°2. Read the following two statements and choose the

correct option.Statement I : Diameter is the longest chord of acircle.Statement II : A circle has only finite number ofequal chords.

(a) only I is true(b) only II is true(c) both I and II are true(d) neither I nor II is true

3. In the given figure, O is the centre of the circle.If OA = 5 cm and OC = 3 cm, then the length ofAB is :

(a) 4 cm (b) 6 cm (c) 8 cm (d) 15 cm4. If one side of a cyclic quadrilateral is produced,

then the exterior angle is equal to its :(a) exterior adjacent angle(b) alternate angle

(c) interior opposite angle(d) corresponding angle

5. Three chords AB, CD and EF of a circle arerespectively 3 cm, 3.5 cm and 3.8 cm away fromthe centre. Then which of the following relationsis correct ?

(a) AB > CD > EF (b) AB < CD < EF(c) AB = CD = EF (d) none of these

6. In the given figure, O is the centre of the circle.If ∠CAB = 40° and ∠CBA = 110°, the value ofx is :

(a) 50° (b) 80° (c) 55° (d) 60°7. In a circle, chord AB of length 6 cm is at a distance of

4 cm from the centre O. The length of another chordCD which is also 4 cm away from the centre is :

(a) 6 cm (b) 4 cm (c) 8 cm (d) 3 cm8. In the figure, chord AB is greater than chord CD.

OL and OM are the perpendiculars from the centreO on these two chords as shown in the figure. Thecorrect releation between OL and OM is :

(a) OL = OM (b) OL < OM(c) OL > OM (d) none of these

l Equal chords of a circle (or of congruent circles)are equidistant from the centre (or centres).

l Chords equidistant from the centre of a circle areequal in length.

l The angle subtended by an arc at the centre is doublethe angle subtended by it at any point on theremaining part of the circle.

l Angles in the same segment of a circle are equal.l Points which lie on the same circle are called

concyclic points.

l A quadrilateral is said to be a cyclic quadrilateral ifthere is a circle passing through all its four vertices.

l If a line segment joining two points subtends equalangles at two other points lying on the same side ofthe line containing the line segment, the four pointslie on a circle (i.e., they are concyclic).

l The sum of either pair of opposite angles of a cyclicquadrilateral is 180°.

l If the sum of a pair of opposite angles of aquadrilateral is 180°, the quadrilateral is cyclic.

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9. Number of circles passing through two givenpoints is :

(a) one (b) two (c) finite (d) infinite10. The length of a chord in a circle of diameter

10 cm is 6 cm. The distance of the chord from itscentre is :

(a) 5 cm (b) 3 cm (c) 8 cm (d) 4 cm11. In the given figure, ABC is

an equilateral triangle. Thenmeasure of ∠BEC is :

(a) 100°(b) 120°(c) 140°(d) 90°

12. Two chords AB and CD subtend x° each at thecentre of the circle. If chord AB = 8 cm, thenchord CD is :

(a) 4 cm (b) 8 cm (c) 16 cm (d) 12 cm13. The radius of a circle is 10 cm and the length of

the chord is 12 cm.The distance of the chord fromthe centre is :

(a) 12 cm (b) 10 cm (c) 8 cm (d) 13 cm14. In the figure, O is the centre of the circle and

∠AOB = 80°. The value of x is :

(a) 30° (b) 40° (c) 60° (d) 160°15. In the given figure, a circle with centre O is shown,

where ON > OM. Then which of the followingrelations is true between the chord AB and chordCD ?

(a) AB = CD (b) AB > CD(c) AB < CD (d) none of these

16. In the figure, if AOB is a diameter of the circleand AC = BC, then ∠CAB is equal to :

(a) 30° (b) 60° (c) 90° (d) 45°

17. AD is a diameter of a circle and AB is a chord.If AD = 34 cm, AB = 30 cm, then the distance ofAB from the centre of the circle is :

(a) 17 cm (b) 15 cm (c) 4 cm (d) 8 cm

18. If AB = 12 cm, BC = 16 cm and AB isperpendicular to BC, then the radius of the circlepassing through the points A, B and C is :

(a) 6 cm (b) 8 cm (c) 10 cm (d) 12 cm

19. In the figure, O is the centre of the circle. If∠OAB = 40°, then ∠ACB is equal to :

(a) 50° (b) 40° (c) 60° (d) 70°20. In the figure , if ∠DAB = 60°, ∠ABD = 50°, then

∠ACB is equal to :

(a) 60° (b) 50° (c) 70° (d) 80°21. In the figure, O is the centre of the circle. If ∠ABC

= 20°, then ∠AOC is equal to :

(a) 20° (b) 40° (c) 60° (d) 10°

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22. In the figure, if ∠SPR = 73°, ∠SRP = 42°, then∠PQR is equal to :

(a) 65° (b) 70° (c) 74° (d) 76°23. In the figure, O is the centre of the circle. If ∠OPQ

= 25° and ∠ORQ = 20°, then the measures of∠POR and ∠PQR respectively are :

(a) 90°, 45° (b) 105°, 450°(c) 110°, 55° (d) 100°, 50°

24. Two circles intersect at the points A and B. ADand AC are diameters of the respsective circles asshown in the following figure. Sum of ∠ABD and∠ABC :

(a) is greater than 180° (b) is equal to 180°(c) is less than 180° (d) has no definite value

25. In the figure, if ∠CAB = 40° and AC = BC, then∠ADB equal to :

(a) 40° (b) 60° (c) 80° (d) 100°

26. In the figure, O is the centre of the circle and∠PQR = 100°. Then the reflex ∠POR is :

(a) 280° (b) 200° (c) 260° (d) none of these27. In the given figure, E is any point in the interior

of the circle with centre O. Chord AB =Chord AC. If ∠OBE = 20°, then the value of xis :

(a) 40° (b) 45° (c) 50° (d) 70°28. In the figure, O is the centre of the circle and

∠AOB = 60°. The value of x is :

(a) 30° (b) 35° (c) 25° (d) 40°29. In the figure, AB and CD are two chords of a

circle with centre O and MN as diameter. Theyintersect at a point E. If ∠AEN = ∠DEN = 45°and AB = 6.5 cm, then the length of chord CD isequal to :

(a) 13 cm (b) 6.5 cm(c) 7.0 cm (d) none of these

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30. In the figure, points A, B, C and D lie on a circle.BC is produced to P and ∠BAD = 100°. Themeasure of ∠DCP is :

(a) 100° (b) 180° (c) 110° (d) 90°31. In the figure, chord DE is parallel to the diameter

AC of the circle. If ∠CBE = 60°, then the measureof ∠CED is :


∠ABP = 40°. The measure of ∠PQB is :

(a) 40° (b) 50° (c) 100° (d) 80°33. In the figure, A, B, C and D are four points on a

circle. AC and BD intersect at a point E. If∠BEC = 140° and ∠ECD = 30°, then the value of∠BAC is :


∠AOC = 130°. The value of x is :

(a) 25° (b) 50° (c) 40° (d) 35°

35. In the figure, O is the centre of the circle of radius5 cm. OP ⊥ AB, OQ ⊥ CD, AB || CD, AB = 8 cmand CD = 6 cm. The length of PQ is :

(a) 8 cm (b) 1 cm(c) 6 cm (d) none of these

36. In the figure, O is the centre of the circle and∠ABD = 45°. The value of x is :

(a) 90° (b) 45°(c) 135° (d) none of these

37. In the figure, O is the centre of the circle and∠AOC = 130°. Then ∠ADC is :

(a) 65° (b) 230°(c) 130° (d) 115°

38. In the figure, ∠AOB = 90° and ∠ABC = 30°,then ∠CAO is equal to :

(a) 30° (b) 45°(c) 90° (d) 60°

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ROTHERS PRAKASHAN

39. In the figure, AB and CD are two equal chords ofa circle with centre O, OP and OQ areperpendiculars on chords AB and CD, respectively.If ∠POQ = 150°, then ∠APQ is equal to :

(a) 30° (b) 75° (c) 15° (d) 60°

40. ABCD is a cyclic quadrilateral such that AB is adiameter of the circle circumscribing it and ∠ADC= 140°, then ∠BAC is equal to :

(a) 80° (b) 50° (c) 40° (d) 30°

41. In the figure, BC is a diameter of the circle and∠BAO = 60°. Then ∠ADC is equal to :

(a) 30° (b) 45° (c) 60° (d) 120°42. In the figure, two congruent circles have centres

O and O′. Arc AXB subtends an angle of 75° atthe centre O and arc A′ Y B′ subtends an angle of25° at the centre O′. Then the ratio of arcs AXBand A′ Y B′ is :

(a) 2 : 1 (b) 1 : 2 (c) 3 : 1 (d) 1 : 3

B. Questions From CBSE Examination Papers

1. In the figure, if ∠ACB = 50°, then ∠OAB is :[T-II (2011)]

(a) 50° (b) 40° (c) 60° (d) 70°2. In the figure, if OA = 17 cm, AB = 30 cm and OD

is perpendicular to AB, then CD is equal to :[T-II (2011)]

(a) 8 cm (b) 9 cm (c) 10 cm (d) 11 cm

3. ABCE is a cyclic quadrilateral. O is the centre ofthe circle and ∠AOC = 150°, then ∠CBD :

[T-II (2011)]

(a) 225° (b) 128° (c) 150° (d) 75°4. Which of the following pairs of angles are opposite

angles of a cyclic quadrilateral? [T-II (2011)](a) 131°, 28° (b) 95°, 55°(c) 123°, 57° (d) 64°, 52°

5. A circle divides a plane in which it lies includingitself in : [T-II (2011)]

(a) 2 parts (b) 3 parts (c) 4 parts (d) 5 parts6. O is the centre of the circle ∠QPS = 65°; ∠PRS

= 33°′, ∠PSQ is : [T-II (2011)]

(a) 90° (b) 82° (c) 102° (d) 42°

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ROTHERS PRAKASHAN

7. In the figure, O is the centre of the circle. ∠POQ= 100°; ∠POR = 110°; then ∠QPR equals :

[T-II (2011)]

(a) 210° (b) 200° (c) 150° (d) 75°8. In the figure ∠CAB = 45° ; ∠DBC = 55°, then

∠DCB equals : [T-II (2011)]

(a) 55° (b) 80° (c) 100° (d) 120°9. In the figure, PQ is the diameter of the semicircle

in which ∠SPR = 30° ; ∠QPR = 20° ; then ∠SRPequals : [T-II (2011)]

(a) 40° (b) 65° (c) 120° (d) 35°10. In the figure, O is the centre of the circle of radius

13 cm and chord AB is of length 24 cm. If OC isperpendicular from the centre to AB, then OCequals : [T-II (2011)]

(a) 26 cm (b) 10 cm (c) 5 cm (d) 8 cm11. In the figure, if O is the centre of the circle;

OL = 4 cm, AB = 6 cm and OM = 3 cm, thenCD is equal to : [T-II (2011)]

(a) 4 cm (b) 8 cm (c) 6 cm (d) 10 cm

12. In the figure, O is the centre of the circle, find∠AOC, given ∠BAO = 30° and ∠BCO = 40°.

[T-II (2011)]

(a) 35° (b) 140°(c) 70° (d) cannot be determined

13. In the figure, O is the centre of a circle and ∠OBA= 60°. Then ∠ACB equals : [T-II (2011)]

(a) 60° (b) 120° (c) 75° (d) 30°

14. In the figure, AOB is a diameter of the circle andAC = BC. Then ∠CAB is : [T-II (2011)]

(a) 30° (b) 45° (c) 60° (d) 90°

15. If AOB is the diameter of the circle and∠B = 35°, then x equals : [T-II (2011)]

(a) 90° (b) 55° (c) 75° (d) 45°

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16. Given two concentric circles with centre O. A linecuts the circles at A, B, C, D, respectively. If AB= 10 cm, then length CD is : [T-II (2011)]

(a) 5 cm (b) 10 cm(c) 7.5 cm (d) none of these

17. In the figure, O is the centre of the circle, ∠CBE= 25° and ∠DEA = 60°. The measure of ∠ADBis : [T-II (2011)]

(a) 90° (b) 95° (c) 85° (d) 120°18. Given three collinear points, then the number of

circles which can be drawn through these pointsis : [T-II (2011)]

(a) zero (b) one (c) two (d) infinite19. The length of chord which is at a distance of

12 cm from centre of circle of radius 13 cm is :[T-II (2011)]

(a) 5 cm (b) 12 cm (c) 13 cm (d) 10 cm20. In the figure, AB is a diameter of the circle.

CD || AB and ∠BAD = 40°, then ∠ACD is :[T-II (2011)]

(a) 40° (b) 90° (c) 130° (d) 140°21. In the figure, the values of x and y respectively

are : [T-II (2011)]

(a) 20°, 30° (b) 36°, 60° (c) 15°, 30° (d) 25°, 30°22. The distance of a chord 8 cm long from the centre

of a circle of radius 5 cm is : [T-II (2011)](a) 4 cm (b) 3 cm (c) 2 cm (d) 9 cm

23. In the given figure, if POQ is a diameter of thecircle and PR = QR, then ∠RPQ is equal to :

[T-II (2011)]

(a) 30° (b) 60° (c) 90° (d) 45°24. For what value of x in the figure, points A, B, C

and D are concyclic? [T-II (2011)]

(a) 9° (b) 10° (c) 11° (d) 12°25. Two circles are said to be concentric, if :

[T-II (2011)](a) they have same radius(b) they have different radii(c) they have same centre(d) their centres are collinear

26. In the figure, if O is the centre, then the value ofy is : [T-II (2011)]

(a) 35° (b) 75° + x (c) 70° – x (d) 140°33. In a circle with centre O, chords AB and CD are

of length 5 cm and 6 cm respectively and subtendangle x° and y° at centre of circle respectivelythen : [T-II (2011)]

(a) x° = y° (b) x° < y°(c) x° > y° (d) none of the above

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28. ABCD is a cyclic quadrilateral as shown in thefigure. The value of (x + y) is : [T-II (2011)]

(a) 200° (b) 100° (c) 180° (d) 160°29. AD is a diameter of a circle and AB is a chord.

If AD = 34 cm and AB = 30 cm, the distance ofAB from the centre of the circle is : [T-II (2011)]

(a) 17 cm (b) 15 cm (c) 4 cm (d) 8 cm30. In the figure, if O is the centre and ∠BOA = 120°,

then the value of x is : [T-II (2011)]

(a) 120° (b) 60° (c) 30° (d) 90°31. If AB = 12 cm, BC = 16 cm and AB ⊥ BC, then

radius of the circle passing through A, B and Cis : [T-II (2011)]

(a) 6 cm (b) 8 cm (c) 10 cm (d) 12 cm32. An equilateral ∆ABC is inscribed in a circle with

centre O. The measure of ∠BOC is : [T-II (2011)](a) 110° (b) 100° (c) 120° (d) 130°

33. In the figure, quadrilateral PQRS is cyclic. If ∠P= 80°, then ∠R is : [T-II (2011)]


∠ABC = 55°, then ∠ADC is : [T-II (2011)]

(a) 55° (b) 110° (c) 75° (d) 27.5°35. In the figure, if ∠PQR = 40°, then the value of

∠PSR is : [T-II (2011)]


∠ABC = 36°. The measure of ∠AOC is :[T-II (2011)]

(a) 36° (b) 72° (c) 144 (d) 18°37. In the figure, if AB is the diameter of the circle,

then the value of x is : [T-II (2011)]

(a) 40° (b) 50° (c) 80° (d) 90°38. In the figure, if ∠OAB = 40°, then ∠ACB is equal

to : [T-II (2011)]

(a) 50° (b) 40° (c) 60° (d) 70°39. In the figure, O is the centre of the circle with

∠AOB = 85° and ∠AOC = 115°, then ∠BACis : [T-II (2011)]

(a) 115° (b) 85° (c) 80° (d) 100°

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ROTHERS PRAKASHAN

1. Find the the length of a chord which is at a dis-tance of 5 cm from the centre of a circle whoseradius is 13 cm.

2. The radius of a circle is 10 cm and a chord of thecircle is 12 cm long. Find the distance of the chordfrom the centre of the circle.

3. Can we have a cyclic quadrialateral ABCDsuch that ∠A = 90°, ∠B = 70°, ∠C = 95° and∠D = 105° ?

4. Two congruent circles with centres O and O′intersect at two points A and B. Check whether∠AOB = ∠AO′ B or not.

5. In the figure, PQR is right angled at Q. Point S istaken on side PR such that PS = SR andQR = QS. Find the measure of ∠QSR.

6. Show that diameter of a circle is the greatest chord.7. In the figure, O is the centre of the circle. If ∠BAC

= 50°, find ∠BOC.

8. In the figure, ∆ABC is equilateral. Find ∠BDCand ∠BEC.

9. Check whether the following statement is true.A, B, C, D are four points such that

∠BAC = 45° and ∠BDC = 45°, then A, B, C, Dare concyclic.

10. In the figure, if AOB is a diameter and∠ADC = 120°, find ∠CAB.

11. In the figure, O is the centre of the circle, BD =DC and ∠DBC = 30°. Find the measure of∠BAC.

12. If arcs AXB and CYD of a circle are congruent,find the ratio of AB and CD.

13. AOB is a diameter of a circle and C is a point onthe circle. Check whether AC2 + BC2 = AB2 istrue or not.

14. Two chords of a circle of lengths 10 cm and 8 cmare at the distances 8 cm and 3.5 cm respectivelyfrom the centre. Check whether the abovestatement is true or not.

SHORT ANSWER TYPE QUESTIONS [2 Marks]

A. Important Questions


1. In the figure, A, B, C,D are four points on acircle. AC and BD in-tersect at a point E suchthat ∠BEC = 130° and∠ECD = 20°. Find∠BAC. [T-II (2011)]

2. In the figure, ∠PQR= 100°, where P, Q, Rare points on a circle,with centre O. Find∠OPR. [T-II (2011)]

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3. In the figure, ∠ABC = 69°; ∠ACB = 31°. Find∠BDC. [T-II (2011)]

4. Prove that equal chords of a circle subtend equalangles at the centre. [T-II (2011)]

5. AB and CD are two parallel chords on the sameside of the circle such that AB = 6 cm;CD = 8 cm. The small chord is at a distance of 4cm from the centre. At what distance from thecentre is the other chord. [T-II (2011)]

6. In ΔABE, AE = BE. Circle through A and Bintersects AE and BE at D and C. Prove thatDC | | AB. [T-II (2011)]

7. In the figure, O is the centre of the circle and∠BOC = 120°, find ∠CDE. [T-II (2011)]

8. In the figure, AB is the diameter of the circle withcentre O. If ∠DAB = 70° and ∠DBC = 30°,determine ∠ABD, ∠CDB. [T-II (2011)]

9. In the figure, OD ⊥ AB and AC is a diameter.Show BC = 2 (OD). [T-II (2011)]

10. Two concentric circles with centre O have A, B,C and D as points of intersection with a line l asshown in the figure, If AD = 12 cm andBC = 8 cm, find the length of AB and CD.

[T-II (2011)]

11. In the figure, O is the centre of the circle. If∠D = 130°, then find ∠CAB. [T-II (2011)]

12. In the figure, ∠PQR = 100°, where P, Q and R arepoints on a circle with centre O. Find ∠OPR.

[T-II (2011)]

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13. ABCD is a cyclic quadrilateral. Find ∠ADB.[T-II (2011)]

14. In the figure, ABCD is a cyclic quadrilateral. AEis drawn parallel to CD and BA is produced upto F. If ∠ABC = 92°, ∠FAE = 20°, find ∠BCD.

[T-II (2011)]

15. ABDC is a cyclic quadrilateral and AB = AC. If∠ACB = 70°, find ∠BDC. [T-II (2011)]

16. In the figure, OA = OB = OC. Show that∠x + ∠y = 2(∠z + ∠t). [T-II (2011)]

17. In the figure, l is a line intersecting two concentriccircles with centre P at points A, C, D and B.Show that AC = DB. [T-II (2011)]

18. Suppose you are given a circle. Give a constructionto find its centre. [T-II (2011)]

19. Prove that equal chords of a circle subtend equalangles at the centre. [T-II (2011)]

20. A chord of a circle is equal to the radius of thecircle. Find the angle subtended by the chord at apoint on the major arc. [T-II (2011)]

21. In the figure, AB and CD are two equal chords ofa circle with centre O. OP and OQ areperpendiculars on chords AB and CD respectively.If ∠POQ = 150°, find ∠APQ. [T-II (2011)]

22. In the figure, chord AB of circle with centre O isproduced to C such that BC = OB. CO is joinedand produced to meet the circle in D. If∠ACD = y and ∠AOD = x, show that x = 3y.

[T-II (2011)]

23. O is the centre of a circle and ∠BOA = 90°, ∠COA= 110°. Find the measure of ∠BAC. [T-II (2011)]

24. If O is centre of circle shown in the figure and∠AOB = 110°, then find ∠BCD. [T-II (2011)]

A

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25. Prove that the line drawn through the centre of acircle to bisect a chord is perpendicular to thechord. [T-II (2011)]

26. Two circles intersect at two points A and B. ADand AC are diameters to the two circles. Provethat B lies on the line segment DC. [T-II (2011)]

27. In the figure, O is the centre of the circle. Theangle subtended by arc ABC at the centre is 140°.AB is produced to P. Determine ∠ADC and ∠CBP.

[T-II (2011)]

28. In the figure, O is the centre of the circle and∠BAC = 60°. Find the value of x. [T-II (2011)]

29. In the figure, ABCD is a cyclic quadrilateral and∠ABC = 85°. Find the measure of ∠ADE.

[T-II (2011)]

30. In the figure, O is the centre of the circle, OM ⊥BC, OL ⊥ AB, ON ⊥ AC and OM = ON = OL.Is ΔABC equilateral? Give reasons.[T-II (2011)]

31. In the figure, ∠ABC = 45°. Prove that OA ⊥ OC.[T-II (2011)]

32. In a cyclic quadrilateral ABCD, if AB | | CD and∠B = 70°, find the measures of the remainingangles of the quadrilateral. [T-II (2011)]

33. Find the length of the chord, which is at a distanceof 3 cm from the centre of a circle of radius5 cm. [T-II (2011)]

34. Find the radius of a chord, which is at a distanceof 4 cm from the centre of a circle whose radiusis 5 cm. [T-II (2011)]

35. In the figure, O is the centre of the circle. If ∠OAC= 35° and ∠OBC = 40°, find the value of x.

[T-II (2011)]

36. O is the circumcentre of the ∆ABC and D is themid point of the base BC. Prove that∠BOD = ∠A. [T-II (2011)]

37. Prove that any cyclic parallelogram is a rectangle.[T-II (2011)]

38. AOB is a diameter of the circle and C, D, E areany three points on the semicircle. Find the valueof ∠ACD + ∠BED. [T-II (2011)]

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39. Two congruent circles intersect each other at pointsA and B. Through A a line segment PAQ is drawnso that P and Q lie on the two circles. Prove thatBP = BQ. [T-II (2011)]

40. Two parallel chords of a circle whose diameter is13 cm are respectively 5 cm and 12 cm. Find thedistance between them if they lie on opposite sidesof centre. [T-II (2011)]

41. In the figure, ABCD is a cyclic quadrilateral. If

∠BCD = 120° and ∠ABD = 50°, find ∠ADB.[T-II (2011)]

SHORT ANSWER TYPE QUESTIONS [3 Marks]

A. Important Questions1. If the perpendicular bisector of a chord AB of a

circle PXAQBY intersects the circle at P and Q,prove that arc PXA ≅ arc PYB.

2. In the figure, AOC is a diameter of the circle and

arc AXB = 12 arc BYC. Find ∠BOC.

3. A quadrilateral ABCD is inscribed in a circle suchthat AB is a diameter and ∠ADC = 130°. Find∠BAC.

4. If two sides of a cyclic quadrilateral are parallel,prove that remaining two sides are equal and bothdiagonals are equal.

5. If a pair of opposite sides of cyclic quadrilateralare equal, prove that its diagonals are also equal.

6. AB and AC are two equal chords of a circle. Provethat the bisector of the angle BAC passes throughthe centre of the circle.

7. B is a point on the minor arc AC of a circle withcentre D. ∠BAC = x° and ∠ADC = y°. Find thevalues of x and y if ABCD is a parallelogram.

8. If a line is drawn parallel to the base of an isosce-les triangle to intersect its equal sides, prove thatthe quadrilateral so formed is cyclic.

9. On a common hypotenuse AB, two right trianglesACB and ADB are situated on opposite sides.Prove that ∠BAC = ∠BDC.

10. In the figure, O is the centre of the circle. Calculate∠APC and ∠AOC.

11. In the figure, AB and BC are two chords of acircle whose centre is O such that ∠ABO =∠CBO. Show that AB = CB.

12. Two circles with centres O and O′ intersect atpoints A and B. A line PQ is drawn parallel toO O′ through A (or B) intersecting the circles at Pand Q. Prove that PQ = 2OO′.

1. An equilateral triangle of side 9 cm is inscribed ina circle. Find its radius. [T-II (2011)]

2. Prove that a cyclic trapezium is always an isosce-les trapezium. [T-II (2011)]

3. If two circles intersect at the two points, provethat their centers lie on the perpendicular bisector

of the common chord. [T-II (2011)]4. In the given figure, AB is a diameter of the circle;

CD is a chord equal to the radius of the circle. ACand BD when extended intersect at a point E. Provethat ∠AEB = 60°. [T-II (2011)]

15

5. If non parallel sides of trapezium are equal, provethat it is cyclic. [T-II (2011)]

6. The radius of a circle is 5 cm and the length of achord in the circle is 8 cm. Find the distance ofthe chord from the centre of the circle.

[T-II (2011)]7. In the figure, if ∠BAC = 60°, ∠ACB = 20°, find

∠ADC. [T-II (2011)]

8. Two circles of radii 10 cm and 8 cm intersect andthe length of the common chord is 12 cm. Findthe distance between their centres. [T-II (2011)]

9. In the figure, O is the centre of the circle ofradius 5 cm. OP ⊥ AB, OQ ⊥ CD, AB || CD. IfAB = 6 cm, CD = 8 cm, determine PQ.

[T-II (2011)]

10. In the figure, ABCD is a parallelogram. The circlethrough A, B and C intersects CD produced at E.Prove that the AE = AD. [T-II (2011)]

1. Show that two circles cannot intersect at morethan two points.

2. Show that the altitudes of a triangle are concur-rent.

3. Prove that angle bisector of any angle of a triangleand the perpendicular bisector of the opposite sideif intersect, they will intersect on the circumcircleof the triangle.

4. If bisectors of opposite angles of a cyclicquadrilateral ABCD intersect the circle,circumscribing it at the points P and Q, prove thatPQ is a diameter of the circle.

5. ABC is an equilateral triangle inscribed in a circleand P be any point on the minor arc BC which

does not coincide with B or C. Prove that PA isangle bisector of ∠BPC.

6. In the figure, O is the centre of the circle. IfBD = OD and CD ⊥ AB, find ∠CAB.

7. Prove that the angles in a segment greater than asemi-circle is less than a right angle.

B. Questions From CBSE Examination Papers1. If two intersecting chords of a circle make equal

angles with the diameter passing through their pointof intersection. Prove that the chords are equal.

[T-II (2011)]2. AB and AC are equal chords of a circle with centre

at O. Show that AO is perpendicular bisector ofBC. [T-II (2011)]

GOYAL B

ROTHERS PRAKASHAN

LONG ANSWER TYPE QUESTIONS [4 Marks]

A. Important Questions

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3. AB and AC are two chords of a circle of radius runits. If AB = 2AC, and the length of theperpendicular from the centre on these chords area and b respectively, prove that 4b2 = a2 + 3r2.

[T-II (2011)]

4. P is the centre of the circle. Prove that∠XPZ = 2 [∠XYZ + ∠XZY]. [T-II (2011)]

5. Prove that the angle subtended by an arc at thecentre is double the angles subtended by it at anypoint on the remaining part of the circle.

[T-II (2011)]

6. If the diagonals of a cyclic quadrilateral arediameters of the circle through the vertices of thequadrilateral, prove that it is a rectangle.

[T-II (2011)]

7. In the figure, ∠ACE = 36°, ∠CAE = 41°. Find x,y and z. [T-II (2011)]

8. In the figure, ABCD is a cyclic quadrilateral, O isthe centre of the circle. If ∠BOD = 160°, find∠BPD. [T-II (2011)]

9. In the figure, ABCD is a cyclic quadrilateral inwhich AB is produced to F and BE || DC. If ∠FBE= 20° and ∠DAB = 95°, find ∠ADC.

[T-II (2011)]

10. In a circle of radius 5 cm, AB and AC are twochords such that AB = AC = 6 cm. Find the lengthof chord BC. [T-II (2011)]

11. In the figure, B and E are points on the linesegment AC and DF respectively. Show thatAD || CF. [T-II (2011)]

12. If O is the centre of a circle as shown in the givenfigure, then prove that x + y = z. [T-II (2011)]

13. Prove that the quadrilateral formed (if possible)by the internal angle bisectors of any quadrilateralis cyclic. [T-II (2011)]

14. Two equal chords AB and CD of a circle whenproduced, intersect at the point P. Prove thatPB = PD. [T-II (2011)]

C

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ROTHERS PRAKASHAN

15. In the figure, equal chords AB and CD intersecteach other at Q at right angle. P and R are midpoints of AB and CD respectively. Show thatOPQR is a square. [T-II (2011)]

16. Prove that line joining the centers of twointersecting circles subtends equal angles at thetwo points of intersection of circles. [T-II (2011)]

17. In the figure, find the values of a, b, c and d.Given ∠BCD = 43° and ∠BAE = 62°.

[T-II (2011)]

18. Two circles intersect at two points B and C.Through B, two line segments ABD and PBQ aredrawn to intersect the circles at A, D and P, Qrespectively. Prove that ∠ACP = ∠QCD.

[T-II (2011)]

19. Prove that the circle drawn on any one of theequal sides of an isosceles triangle as diameterbisects the base of the triangle. [T-II (2011)]

FORMATIVE ASSESSMENT

Activity-1Objective : To verify that the angle subtended by an arc at the centre of a circle is twice the angle subtended by

the same arc at any other point on the remaining part of the circle, using the method of paper cutting,pasting and folding.

Materials Required : White sheets of paper, tracing paper, a pair of scissors, gluestick, colour pencils, geometrybox, etc.

Procedure :1. On a white sheet of paper, draw a circle of any convenient radius with centre O. Mark two points A and

B on the boundary of the circle to get arc AB. Colour the minor arc AB green.

Figure-1

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ROTHERS PRAKASHAN

2. Take any point P on the remaining part of the circle. Join OA, OB, PA and PB.

Figure-2

3. Make two replicas of ∠APB using tracing paper. Shade the angles using different colours.

Figure-3

4. Paste the two replicas of ∠APB adjacent to each other on ∠AOB as shown in the figure.

Figure-4Observations :

1. In figure 2, ∠AOB is the angle subtended by arc AB at the centre and ∠APB is the angle subtended by arcAB on the remaining part of the circle.

2. In figure 3, each angle is a replica of ∠APB.3. In figure 4, we see that the two replicas of ∠APB completely cover the angle AOB.

So, ∠AOB = 2∠APB.Conclusion : From the above activity, it is verified that the angle subtended by an arc at the centre of a circle istwice the angle subtended by the same arc at any other point on the remaining part of the circle.Do Yourself : Verify the above property by taking three circles of different radii.

19

GOYAL B

ROTHERS PRAKASHAN

Activity-2

Objective : To verify that the angles in the same segment of a circle are equal, using the method of paper cutting,pasting and folding.

Materials Required : White sheets of paper, tracing paper, a pair of scissors, gluestick, colour pencils, geometry box,etc.

Procedure :1. On a white sheet of paper, draw a circle of any convenient radius. Draw a chord AB of the circle.

Figure-1

2. Take any three points P, Q and R on the major arc AB of the circle. Join A to P, B to P, A to Q, B to Q, A to Rand B to R.

Figure-2

3. On a tracing paper, trace each of the angles APB, AQB and ARB. Shade the traced copies using differentcolours.

Figure-3

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ROTHERS PRAKASHAN

4. Place the three cut outs one over the other such that the vertices P, Q and R coincide and PA, QA and RA fallalong the same direction.

Figure-4

Observations :1. In figure 2, ∠APB, ∠AQB and ∠ARB are the angles in the same major segment AB.2. In figure 4, we see that ∠APB, ∠AQB and ∠ARB coincide.

So, ∠APB = ∠AQB = ∠ARBConclusion : From the above activity, it is verified that the angles in the same segment of a circle are equal.Do Yourself : Verify the above property by taking three circles of different radii.

Activity-3Objective : To verify using the method of paper cuting, pasting and folding that

(a) the angle in a semi circle is a right angle(b) the angle in a major segment is acute(c) the angle in a minor segment is obtuse.

Materials Required : White sheets of paper, tracing paper, cut out of a right angle, colour pencils, a pair of scissors,gluestick, geometry box, etc.

Procedure : (a) To verify that the angle in a semicircle is a right angle :1. On a white sheet of paper, draw a circle of any convenient radius with centre O. Draw its diameter AB as

shown.

Figure-1

2. Take any point P on the semicircle. Join A to P and B to P.

Figure-2

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ROTHERS PRAKASHAN

3. Make two replicas of ∠APB on tracing paper. Shade the replicas using different colours.

Figure-34. On a white sheet of paper, draw a straight line XY. Paste the replicas obtained in figure 3 on

XY and adjacent to each other such that AP and BP coincide as shown in the figure.

Figure-4(b) To verify that the angle in a major segment is acute :1. On a white sheet of paper, draw a circle of any convenient radius with centre O. Draw a chord AB which does

not pass through O.

Figure 5

2. Take any point P on the major segment. Join P to A and P to B.

Figure-6

3. Trace ∠APB on a tracing paper.

Figure-7

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ROTHERS PRAKASHAN

4. Paste the traced copy of ∠APB on the cut out of a right angled triangle XYZ, right-angled at Y such that PAfalls along YZ.

Figure-8

(c) To verify that the angle in a minor segment is obtuse :1. On a white sheet of paper, draw a circle of any convenient radius with centre O. Draw any chord AB which

does not pass through O.

Figure-9

2. Take any point P on the minor segment. Join P to A and P to B.

Figure-10

3. Trace ∠APB on a tracing paper.

Figure-11

4. Paste the traced copy of ∠APB on the cut out of a right-angled triangle XYZ, right angled at Y, such that PAfalls along YZ.

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GOYAL B

ROTHERS PRAKASHAN

Figure-12Observations :

1. In figure 2, APB is a semicircle. So, ∠APB is an angle in a semicircle.2. In figure 4, we see that PB and PA fall along XY.

Or ∠APB + ∠APB = a straight angle = 180°⇒ 2∠APB = 180°⇒ ∠APB = 90°Hence, angle in a semicircle is a right angle.

3. In figure 7, ∠APB is an angle formed in the major segment of a circle.4. In figure 8, we see that the side PB of ∠APB lies to the right of XY of ∠XYZ, ie, ∠APB is less than a right

angle, or ∠ΑPB is acute.Hence, the angle in a major segment is acute.

5. In figure 11, ∠APB is an angle formed in the minor segment of a circle.6. In figure 12, we see that the side PB of ∠PAB lies to the left of XY of ∠XYZ ie, ∠APB is greater than ∠XYZ

or ∠APB is obtuse.Hence, the angle in a minor segment is obtuse.

Conclusion : From the above activity, it is verified that :(a) the angle in a semicircle is a right angle.(b) the angle in a major segment is acute.(c) the angle in a minor segment is obtuse.

Activity-4

Objective : To verify using the method of paper cutting, pasting and folding that(a) the sum of either pair of opposite angles of a cyclic quadrilateral is 180°(b) in a cyclic quadrilateral the exterior angle is equal to the interior opposite angle.

Materials Required : White sheets of paper, tracing paper, colour pencils, a pair of scissors, gluestick, geometry box, etc.Procedure :(a) 1. On a white sheet of paper, draw a circle of any convenient radius. Mark four points P, Q, R, S on the circumference

of the circle. Join P to Q, Q to R, R to S and S to P.

Figure-1

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GOYAL B

ROTHERS PRAKASHAN

2. Colour the quadrilateral PQRS as shown in the figure and cut it into four parts such that each part contains oneangle, ie, ∠P, ∠Q, ∠R and ∠S.

Figure-2

3. On a white sheet of paper, paste ∠P and ∠R adjacent to each other. Similarly, paste ∠Q and ∠S adjacent toeach other.

Figure-3

(b) 1. Repeat step 1 of part (a).2. Extend PQ to PT to form an exterior angle RQT. Shade ∠RQT.

Figure-4

3. Trace ∠PSR on a tracing paper and colour it.

Figure-5

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GOYAL B

ROTHERS PRAKASHAN

4. Paste the traced copy of ∠PSR on ∠RQT such that S falls at Q and SP falls along QT.

Figure-6

Observations :1. In figure 2, ∠P, ∠Q, ∠R and ∠S are the four angles of the cyclic quadrilateral PQRS.2. In figure 3(a), we see that ∠R and ∠P form a straight angle and in figure 3(b), ∠Q and ∠S form a straight

angle.So, ∠P + ∠R = 180° and ∠Q + ∠S = 180°.Hence, the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

3. In figure 5, ∠PSR is the angle opposite to the exterior angle RQT.4. In figure 6, we see that ∠PSR completely covers ∠TQR.

Hence, in a cyclic quadrilateral the exterior angle is equal to the interior opposite angle.Conclusion : From the above activity, it is verified that

(a) the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.(b) in a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

assignments in mathematics class ix (term 2) 10. circles · l the collection of all the points in a...

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