# SELINA Solutions for Class 10 Maths Chapter 17 - Circles

## Chapter 17 - Circles Exercise Ex. 17(A)

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

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

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

(ii) Find ∠ACB.

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

(i) OCA ; (ii) OAC.

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

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

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. O_{1} and O_{2} are the centres of two circles.

In the figure, given below, find:

Show steps of your working.

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

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.

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

(i) ADB, (ii) AEB.

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

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

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

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

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

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.

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.

In the following figure,

(i) if BAD = 96^{o}, find BCD and BFE.

(ii) Prove that AD is parallel to FE.

ABCD is a parallelogram. A circle through vertices A and B meets side BC at point P and side AD at point Q. Show that quadrilateral PCDQ is cyclic.

▭ABPQ is a cyclic quadrilateral.

⟹∠A = ∠P …..(Exterior angle property of cyclic quadrilateral) …(1)

▭ABCD is a parallelogram.

⟹ ∠A = ∠C …..(Opposite angles of a parallelogram) ….. (2)

From (1) and (2),

∠P = ∠C…..….(3)

But ∠C + ∠D = 180° …. (Sum of interior angles of a parallelogram is 180°)

From (3), we get

∠P + ∠D = 180°

⟹ PCDQ is a cyclic quadrilateral.

Prove that:

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

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

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

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.

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.

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

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.

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.

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.

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.

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

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

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.

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.

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.

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

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.

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.

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.

(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 = - =

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.

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.

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:

Calculate the angles x, y and z if:

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

(i) Angle ABC

(ii) Angle BEC.

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.

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.

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

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

Use the given figure to find

.

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:

.

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

(i) ∠DAB (ii) ∠DBA

- ABCD is a cyclic quadrilateral

m∠DAB = 180° - ∠DCB

= 180° - 130°

= 50°

- In ∆ADB,

m∠DAB + m∠ADB + m∠DBA = 180°

⇒50° + 90° + m∠DBA = 180°

⇒m∠DBA = 40°

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

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

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

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

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

Hence, show that AC is a diameter.

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

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

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,

## Chapter 17 - Circles Exercise Ex. 17(B)

In a cyclic-trapezium, the non-parallel sides are equal and the diagonals are also equal.

Prove it.

In the following figure, AD is the diameter of the circle with centre O. chords AB, BC and CD are equal. If _{}DEF = 110_{}^{, }calculate :

(i) _{}AFE,(ii) _{}FAB.

_{}

If two sides of a cycli-quadrilateral are parallel; prove thet:

(i) its other two side are equal.

(ii) its diagonals are equal.

The given figure show a circle with centre O. also, PQ = QR = RS and _{}PTS = 75°. Calculate:

(i) _{}POS,

(ii) _{}QOR,

(iii) _{}PQR.

In the given figure, AB is a side of a regular six-sided polygon and AC is a side of a regular eight-sided polygon inscribed in the circle with centre O. calculate the sizes of :

(i) _{}AOB,

(ii) _{}ACB,

(iii) _{}ABC.

In a regular pentagon ABCDE, inscribed in a circle; find ratio between angle EDA and angel ADC.

_{}

In the given figure. AB = BC = CD and _{}ABC = 132° calculate:

(i) _{}AEB,

(ii) _{}AED,

(iii) _{}COD.

In the figure, O is the centre of the circle and the length of arc AB is twice the length of arc BC. If angle AOB = 108° find :

(i) _{}CAB,

(ii) _{}ADB.

The figure shows a circle with centre O. AB is the side of regular pentagon and AC is the side of regular hexagon. Find the angles of triangle ABC.

_{}

In the given figire, BD is a side of a regularhexagon, DC is a side of a regular pentagon and AD is adiameter. Calculate:

(i) _{}ADC,

(ii) _{}BDA,

(iii) _{}ABC,

(iv) _{}AEC.

## Chapter 17 - Circles Exercise Ex. 17(C)

In the given circle with diametre AB, find the valuv of x.

In the given figure, ABC is a triangle in which BAC = 30. Show that BC is equal to the radius of the circum-circle of the triangle ABC, whose centre is O.

Prove that the circle drawn on any one a the equalside of an isoscele triangle as diameter bisects the base.

In the given figure, chord ED is parallel to diameter AC of the circle. Given CBE =, calculateDEC.

The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic. Prove it.

In the figure, ∠DBC = 58°. BD is a diameter of the circle. Calculate :

(i) ∠BDC

(ii) ∠BEC

(iii) ∠BAC

D and E are points on equal sides AB and AC of an isosceles triangle ABC such that AD = AE. Provet that the points B, C, E and D are concyclic.

In the given rigure, ABCD is a cyclic eqadrilateral. AF is drawn parallel to CB and DA is produced to point E. If ADC =,FAE =; determineBCD. Given reason in support of your answer.

If I is the incentre of triangle ABC and AI when produced meets the cicrumcircle of triangle ABC in points D. ifBAC =and ABC =. calculate:

(i)DBC (ii)IBC (iii)BIC.

In the given figure, AB = AD = DC = PB and DBC = x^{o}. Determine, in terms of x :

(i)ABD, (ii)APB.

Hence or otherwise, prove thet AP is parallel to DB.

In the given figure; ABC, AEQ and CEP are straight lines. Show thatAPE andCQE are supplementary.

In the given, AB is the diameter of the circle with centre O.

IfADC = , find angle BOC.

In a cyclic-quadrilateral PQRS, angle PQR =. Sides SP and RQ prouduced meet at point A: whereas sides PQ and SR produced meet at point B.

IfA : B =2 : 1 ; find angles A and B.

In the following figure, ABCD is a cyclic quadrilateral in which AD is parallel to BC.

If the bisector of angle A meet BC at point E and the given circle at point F, prove that:

(i) EF = FC (ii) BF = DF

ABCD is a cyclic quadrilateral. Sides AB and DC produced meet at point E; whereas sides BC and AD produced meet at point F.

IfDCF : F : E = 3 : 5 : 4, find the angles of the cyclic quadrilateral ABCD.

The following figure shows a cicrcle with PR as its diameter. If PQ = 7 cm and QR = 3RS = 6 cm, Find the perimetre of the cyclic quadrilateral PQRS.

In the following figure, AB is the diameter of a circle with centre O. If chord AC = chord AD ,prove that:

(i) arc BC = arc DB

(ii) AB is bisector ofCAD.

Further if the lenghof arc AC is twice the lengthof arc BC find : (a)BAC (b) ABC

In cyclic quadrilateral ABCD; AD = BC,BAC= andCBD= ; find ;

(i)BCD (ii)BCA

(iii)ABC (iv)ADC

In the given figure, ACE = and CAF=; find the values of a, b and c.

In the given figure, AB is parallel to DC ,BCE = and BAC =

Find

(i)CAD (ii)CBD (iii)ADC

ABCD is a cyclic quadrilalteral of a circle with centre O such that AB is a diameter of this circle and the length of the chord CD is equal to the radius of the circle,.if AD and BC produced meet at P, show that APB =.

In the figure, given alongside, CP bisects angle ACB.

Show that DP bisects angle ADB.

In the figure, given below , AD = BC, BAC = and CBD = find:

(i)BCD

(ii)BCA

(iii)ABC

(iv)ADB

In the given figure, AD is a diameter. O is the centre of the circle. AD is parallel to BC and ∠CBD = 32°.

Find:

i. ∠OBD

ii. ∠AOB

iii. ∠BED

i. AD is parallel to BC, i.e., OD is parallel to BC and BD is transversal.