Chapter 17 - Circles Exercise Ex. 17(A)

Question 1

In the given figure, O is the centre of the circle. respectively. Find angle AOC Show your steps of working.

Solution 1

Question 2

In the given figure, ∠BAD = 65°, ∠ABD = 70°, ∠BDC = 45°

(i) Prove that AC is a diameter of the circle.

(ii) Find ∠ACB.

 

 

Solution 2

Question 3

Given O is the centre of the circle and AOB = 70^o.Calculate the value of:

(i) OCA ; (ii) OAC.

Solution 3

Question 4

In each of the following figures, O is the centre of the circle. Find the values of a, b and c.

Solution 4

Question 5

In each of the following figures, O is the centre of the circle. Find the values of a, b, c and d.

Solution 5

Question 6

In the figure, AB is common chord of the two circles. If AC and AD are diameters; prove that D, B and C are in a straight line. O₁ and O₂ are the centres of two circles.

Solution 6

Question 7

In the figure, given below, find:

 Show steps of your working.

Solution 7

Question 8

In the figure, given below, O is the centre of the circle. If

Solution 8

Question 9

Solution 9

Question 10

In the figure given below, ABCD is a cyclic quadrilateral in which BAD= 75^o ; ABD= 58^o and ADC = 77^o. Find:

(i) BDC, (ii) BCD, (iii) BCA.

Solution 10

Question 11

In the figure given below, O is the centre of the circle and triangle ABC is equilateral. Find:

(i) ADB, (ii) AEB.

Solution 11

Question 12

Given CAB = 75^o and CBA = 50^o. Find the value of DAB + ABD.

Solution 12

Question 13

ABCD is a cyclic quadrilateral in a circle with centre O. IfADC = 130^o, find BAC.

Solution 13

Question 14

In the figure given alongside, AOB is a diameter of the circle and AOC = 110^o, find BDC.

Solution 14

Question 15

In the following figure, O is the centre of the circle;AOB = 60^o and BDC = 100^o, find OBC.

Solution 15

Question 16

In ABCD is a cyclic quadrilateral in whichDAC = 27^o,DBA = 50^o and ADB = 33^o. Calculate (i) DBC, (ii) DCB, (iii)CAB.

Solution 16

Question 17

In the figure given alongside, AB and CD are straight lines through the centre O of a circle. If AOC = 80^o and CDE = 40^o . Find the number of degrees in: (i) DCE; (ii) ABC.

Solution 17

Question 18

In the figure given below, AC is a diameter of a circle, whose centre is O. A circle is described on AO as diameter. AE, a chord of the larger circle, intersects the smaller circle at B. Prove that AB = BE.

Solution 18

Question 19

In the following figure,

(i) if BAD = 96^o, find BCD and BFE.

(ii) Prove that AD is parallel to FE.

Solution 19

Question 20

Prove that:

(i) the parallelogram, inscribed in a circle, is a rectangle.

(ii) the rhombus, inscribed in a circle, is a square.

Solution 20

Question 21

In the following figure, AB = AC. Prove that DECB is an isosceles trapezium.

Solution 21

Question 22

Two circles intersect at P and Q. Through P diameters PA and PB of the two circles are drawn. Show that the points A, Q and B are collinear.

Solution 22

Question 23

The figure given below, shows a circle with centre O. Given: AOC = a and ABC = b.

(i) Find the relationship between a and b

(ii) Find the measure of angle OAB, if OABC is a parallelogram.

Solution 23

Question 24

Two chords AB and CD intersect at P inside the circle. Prove that the sum of the angles subtended by the arcs AC and BD as the centre O is equal to twice the angle APC

Solution 24

 

Question 25

In the figure given RS is a diameter of the circle. NM is parallel to RS and MRS = 29^o

Calculate : (i) RNM ; (ii) NRM.

Solution 25

Question 26

In the figure given alongside, AB || CD and O is the centre of the circle. If ADC = 25^o; find the angle AEB. Give reasons in support of your answer.

Solution 26

Question 27

Two circles intersect at P and Q. Through P, a straight line APB is drawn to meet the circles in A and B. Through Q, a straight line is drawn to meet the circles at C and D. Prove that AC is parallel to BD.

Solution 27

Question 28

ABCD is a cyclic quadrilateral in which AB and DC on being produced, meet at P such that PA = PD. Prove that AD is parallel to BC.

Solution 28

Question 29

AB is a diameter of the circle APBR as shown in the figure. APQ and RBQ are straight lines. Find:

Solution 29

Question 30

In the given figure, SP is the bisector of angle RPT and PQRS is a cyclic quadrilateral. Prove that: SQ = SR.

Solution 30

Question 31

In the figure, O is the centre of the circle, AOE = 150^o, DAO = 51^o. Calculate the sizes of the angles CEB and OCE.

Solution 31

Question 32

In the figure, P and Q are the centres of two circles intersecting at B and C. ACD is a straight line. Calculate the numerical value of x.

Solution 32

Question 33

The figure shows two circles which intersect at A and B. The centre of the smaller circle is O and lies on the circumference of the larger circle. Given that APB = a^o. Calculate, in terms of a^o, the value of:

Give reasons for your answers clearly.

Solution 33

Question 34

In the given figure, O is the centre of the circle and ABC = 55^o. Calculate the values of x and y.

Solution 34

Question 35

In the given figure, A is the centre of the circle, ABCD is a parallelogram and CDE is a straight line. Prove that BCD = 2ABE.

Solution 35

Question 36

ABCD is a cyclic quadrilateral in which AB is parallel to DC and AB is a diameter of the circle. Given BED = 65^o; calculate: (i) DAB , (ii) BDC.

Solution 36

Question 37

In the given figure, AB is a diameter of the circle. Chord ED is parallel to AB and EAB = 63^o; calculate: (i) EBA , (ii)BCD.

Solution 37

(i) AEB = 

(Angle in a semicircle is a right angle)

Therefore  EBA =  - EAB =  - = 

(ii) AB  ED

Therefore  DEB = EBA =                         (Alternate angles)

Therefore BCDE is a cyclic quadrilateral

Therefore  DEB  + BCD = 

[Pair of opposite angles in a cyclic quadrilateral are supplementary]

Therefore  BCD =  -  = 

Question 38

In the given figure, AB is a diameter of the circle with centre O. DO is parallel to CB and DCB = 120^o; calculate:

(i) DAB, (ii) DBA, (iii) DBC, (iv) ADC.

Also, show that the AOD is an equilateral triangle.

Solution 38

Question 39

In the given figure, I is the incentre of the ABC. BI when produced meets the circumcirle of ABC at D. Given BAC = 55° and ACB = 65^o ; calculate: (i) DCA, (ii) DAC, (iii) DCI, (iv) AIC.

Solution 39

Question 40

A triangle ABC is inscribed in a circle. The bisectors of angles BAC, ABC and ACB meet the circumcircle of the triangle at points P, Q and R respectively. Prove that:

Solution 40

Question 41

Calculate the angles x, y and z if: 

Solution 41

Question 42

In the given figure, AB = AC = CD and ADC = 38^o. Calculate:

(i) Angle ABC

(ii) Angle BEC.

Solution 42

Question 43

In the given figure, AC is the diameter of circle, centre O. Chord BD is perpendicular to AC. Write down the angles p, q and r in terms of x.

Solution 43

Question 44

In the given figure, AC is the diameter of circle, centre O. CD and BE are parallel. Angle AOB = 80^o and angle ACE = 10^o. Calculate:

(i) Angle BEC ; (ii) Angle BCD ; (iii) Angle CED.

Solution 44

Question 45

In the given figure, AE is the diameter of circle. Write down the numerical value of . Give reasons for your answer.

Solution 45

Question 46

In the given figure, AOC is a diameter and AC is parallel to ED. If , calculate .

Solution 46

Question 47

Use the given figure to find

.

Solution 47

Question 48

In the given figure, AOB is a diameter and DC is parallel to AB. If CAB = x^o ; find (in terms of x) the values of:

.

Solution 48

Question 49

In the given figure, AB is the diameter of a circle with centre O. ∠BCD = 130°. Find:

(i) ∠DAB  (ii) ∠DBA

Solution 49

1. ABCD is a cyclic quadrilateral

m∠DAB = 180° - ∠DCB

= 180° - 130°

= 50°

2. In ∆ADB,

m∠DAB + m∠ADB + m∠DBA = 180°

⇒50° + 90° + m∠DBA = 180°

⇒m∠DBA = 40°

Question 50

In the given figure, PQ is the diameter of the circle whose centre is O. GivenROS = ; calculate RTS.

Solution 50

Question 51

In the given figure, PQ is a diameter. Chord SR is parallel to PQ. Given that PQR = ; calculate

Solution 51

Question 52

AB is the diameter of the circle with centre O. OD is parallel to BC and AOD = ; calculate the numerical values of:

Solution 52

Question 53

In the given figure, the centre O of the small circle lies on the circumference of the bigger circle. If APB =  and BCD = ; find:

Solution 53

Question 54

In the given figure, BAD =  , ABD =  and BDC = ; find:

Hence, show that AC is a diameter.

Solution 54

Question 55

In a cyclic quadrilateral ABCD, A :C = 3 : 1 and B : D = 1 : 5; find each angle of the quadrilateral.

Solution 55

Question 56

The given figure shows a circle with centre O andABP =. Calculate the measure of

Solution 56

Question 57

In the given figure, M is the centre of the circle. Chords AB and CD are perpendicular to each other. If MAD =x and BAC = y,

Solution 57