#### Chapters

Chapter 2: Banking (Recurring Deposit Account)

Chapter 3: Shares and Dividend

Chapter 4: Linear Inequations (In one variable)

Chapter 5: Quadratic Equations

Chapter 6: Solving (simple) Problems (Based on Quadratic Equations)

Chapter 7: Ratio and Proportion (Including Properties and Uses)

Chapter 8: Remainder and Factor Theorems

Chapter 9: Matrices

Chapter 10: Arithmetic Progression

Chapter 11: Geometric Progression

Chapter 12: Reflection

Chapter 13: Section and Mid-Point Formula

Chapter 14: Equation of a Line

Chapter 15: Similarity (With Applications to Maps and Models)

Chapter 16: Loci (Locus and Its Constructions)

Chapter 17: Circles

Chapter 18: Tangents and Intersecting Chords

Chapter 19: Constructions (Circles)

Chapter 20: Cylinder, Cone and Sphere

Chapter 21: Trigonometrical Identities

Chapter 22: Height and Distances

Chapter 23: Graphical Representation

Chapter 24: Measure of Central Tendency(Mean, Median, Quartiles and Mode)

Chapter 25: Probability

## Chapter 17: Circles

#### Exercise 17(A) [Pages 257 - 262]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 17 Circles Exercise 17(A) [Pages 257 - 262]

In the given figure, O is the centre of the circle. ∠OAB and ∠OCB are 30° and 40°

respectively. Find ∠AOC . Show your steps of working.

In the 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°. Calculate the value of:

(i) ∠OCA .

(ii) ∠OCA .

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

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

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

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

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

.

In 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 circle. 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: ∠BCD, Show steps of your working .

In the figure, given below, find: ∠ ADC show steps of your working.

In the figure, given below, find: ∠ ABC Show steps of your working.

In the given figure, O is the centre of the circle. If ∠AOB = 140° and ∠OAC = 50°; Find:

(i) ∠ACB, (ii) ∠OBC, (iii) ∠OAB, (iv) ∠CBA.

Calculate:

(i) ∠CDB, (ii) ∠ABC, (iii) ∠ACB

In the figure, given below, ABCD is a cyclic quadrilateral in which ∠BAD = 75°; ∠ABD = 58°

and ∠ADC = 77°. Find:

(i) ∠BDC, (ii) ∠BCD, (iii) ∠BCA.

In the following figure, O is centre of the circle and ΔABC is equilateral.

Find: (i) ∠ADB, (ii) ∠AEB.

Given: ∠CAB = 75° and ∠CBA = 50°. Find the value of ∠DAB + ∠ABD.

ABCD is a cyclic quadrilateral in a circle with centre O. If ∠ADC = 130°; find ∠ BAC.

In the figure, given below, AOB is a diameter of the circle and ∠AOC = 110°. Find ∠BDC.

In the following figure, O is the centre of the circle,

∠AOB = 60° and ∠BDC = 100° Find ∠OBC.

ABCD is a cyclic quadrilateral in which ∠DAC = 27° ; ∠DBA = 50° and ∠ADB = 33°. Calculate : ∠DBC,

In ABCD is a cyclic quadrilateral in which ∠ DAC = 27° , ∠ DBA = 50° and ∠ ADB = 33°. Calculate ∠ DCB.

In ABCD is a cyclic quadrilateral in which ∠ DAC = 27° , ∠ DBA = 50° and ∠ ADB = 33°. Calculate ∠ CAB.

In the figure given below, AB and CD are straight lines through the centre O of a circle. If ∠AOC

= 80° and ∠CDE = 40°, Find the number of degrees in:

∠ DCE .

In the figure given alongside, AB and CD are straight lines through the centre O of a circle. If ∠AOC = 80° and ∠CDE = 40°, Find the number of degrees in : ∠ ABC .

In the given figure, 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,

If ∠BAD = 96°, find ∠BCD and ∠BFE.

In the following figure, 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.

Prove that:

the parallelogram, inscribed in a circle, is a rectangle.

Prove that:

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 diameter 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.

Find the relationship between a and b

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

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 at the centre O is equal to twice the angle APC.

In the given figure, RS is a diameter of the circle. NM is parallel to RS and ∠MRS = 29°.

Calculate: ∠RNM,

In the given figure, RS is a diameter of the circle. NM is parallel to RS and ∠MRS = 29°.

Calculate : ∠ NRM

In the figure, given alongside, AB ∥ CD and O is the centre of the circle. If ∠ADC = 25°; find

the angle AEB give reasons in support of your answer.

Two circle 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 : ∠PRB

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

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

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

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

In the figure, given below, 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°.

Calculate, in terms of a°, the value of : obtuse ∠AOB,

Give reasons for your answers clearly.

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: ∠ ACB .

Give reasons for your answers clearly.

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. iven that ∠ APB = a^{o}. Calculate, in terms of a^{o}, the value of: ∠ ADB.

Give reasons for your answers clearly.

In the given figure, O is the centre of the circle and ∠ABC = 55°. 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 = 2∠ABE .

ABCD is a cyclic quadrilateral in which AB is parallel to DC and AB is a diameter of the circle. Given ∠BED = 65°; 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°. Calculate : ∠ EBA

In the given figure, AB is a diameter of the circle. Chord ED is parallel to AB and ∠EAB = 63°. Calculate : ∠ BCD.

In the given figure, AB is a diameter of the circle with centre O. DO is parallel to CB and ∠DCB = 120°.

Calculate : ∠DBA

Also, show that the Δ AOD is an equilateral triangle.

In the given figure, AB is a diameter of the circle with centre O. DO is parallel to CB and ∠DCB = 120°.

Calculate : ∠DBC

Also, show that the Δ AOD is an equilateral triangle.

In the given figure, AB is a diameter of the circle with centre O. DO is parallel to CB and ∠DCB = 120°.

Calculate : ∠ADC

Also, show that the Δ AOD is an equilateral triangle.

In the given figure, I is the incentre of ΔABC. BI when produced meets the circumcircle of

ΔABC at D. ∠BAC = 55° and ∠ACB = 65°; 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 :

∠ABC = 2∠APQ,

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:

∠ ACB = 2 ∠APR ,

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:

`angle "QPR" = 90^circ - 1/2 angle "BAC"`

Calculate the angles x, y and z if:

`x /3= y/4 = z/5 `

In the given figure, AB = AC = CD and ∠ADC = 38°. Calculate:

(i) Angle ABC

(ii) Angle BEC

In the given figure, AC is a 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 the circle with centre O. CD and BE are parallel. Angle ∠AOB = 80° and ∠ACE = 10°.

Calculate : Angle BEC,

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 : Angle BCD

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: Angle CED.

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

In the given figure, AOC is a diameter and AC is parallel to ED. If ∠CBE = 64°, Calculate ∠DEC .

Use the given figure to find:

(i) ∠BAD,

(ii) ∠DQB

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

∠COB,

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: ∠ DOC.

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: ∠ DAC

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: ∠ ADC.

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

(i) ∠DAB

(ii) ∠DBA

In the given figure, PQ is the diameter of the circle whose centre is O. Given ∠ROS = 42°, Calculate ∠RTS.

In the given figure, PQ is a diameter. Chord SR is parallel to PQ. Given that ∠PQR = 58°,

Calculate:

(i) ∠RPQ, (ii) ∠STP

AB is the diameter of the circle with centre O. OD is parallel to BC and ∠AOD = 60°. Calculate the numerical values of : ∠ABD

AB is the diameter of the circle with centre O. OD is parallel to BC and ∠ AOD = 60° ; calculate the numerical values of: ∠ DBC

AB is the diameter of the circle with centre O. OD is parallel to BC and ∠ AOD = 60° ; calculate the numerical values of: ∠ ADC

In the given figure, the centre O of the small circle lies on the circumference of the bigger circle. If ∠APB = 75° and ∠BCD = 40°, find : ∠AOB

In the given figure, the centre O of the small circle lies on the circumference of the bigger circle. If ∠APB = 75° and ∠BCD = 40°, find : ∠ACB

In the given figure, the centre O of the small circle lies on the circumference of the bigger circle. If ∠APB = 75° and ∠BCD = 40°, find : ∠ABD

In the given figure, the centre O of the small circle lies on the circumference of the bigger circle. If ∠APB = 75° and ∠BCD = 40°, find : ∠ADB

In the given figure, ∠BAD = 65°, ∠ABD = 70° and ∠BDC = 45°. Find: ∠BCD

Hence, show that AC is a diameter

In the given figure, ∠BAD = 65°, ∠ABD = 70° and ∠BDC = 45°. Find: ∠ ACB.

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 and ∠ABP = 42°

Calculate the measure of:

(i) ∠PQB

(ii) ∠QPB + ∠PBQ

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 : express ∠AMD in terms of x.

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 : express ∠ABD in terms of y.

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 , Prove that : x = y

#### Exercise 17(B) [Page 265]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 17 Circles Exercise 17(B) [Page 265]

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: ∠AEF

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: ∠ FAB.

If two sides of a cyclic quadrilateral are parallel; prove that:

(i) its other two sides are equal.

(ii) its diagonals are equal.

The given figure shows 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 angle ADC.

In the given figure, AB = BC = CD and ∠ABC = 132° . Calcualte: ∠AEB

In the given figure, AB = BC = CD and ∠ABC = 132° . Calcualte: ∠AED

In the given figure, AB = BC = CD and ∠ABC = 132° . Calcualte: ∠ 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 : ∠CAB

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 : ∠ 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 figure, BD is a side of a regular hexagon, DC is a side of a regular pentagon and AD is a diameter. Calculate :

(i) ∠ADC (ii) ∠BDA, (iii) ∠ABC, (iv) ∠AEC.

#### Exercise 17(C) [Pages 265 - 267]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 17 Circles Exercise 17(C) [Pages 265 - 267]

In the given circle with diameter AB, find the value of x.

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

Prove that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base.

In the given figure, chord ED is parallel to diameter AC of the circle. Given ∠CBE = 65°, calculate ∠DEC.

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 : ∠BDC

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

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

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

If I is the incentre of triangle ABC and AI when produced meets the circumcircle of triangle ABC in point D. If ∠BAC = 66° and ∠ABC = 80°. Calculate : ∠DBC

If I is the incentre of triangle ABC and AI when produced meets the cicrumcircle of triangle ABC in points D . if ∠BAC = 66° and ∠ABC = 80°. Calculate : ∠IBC

If I is the incentre of triangle ABC and AI when produced meets the cicrumcircle of triangle ABC in points D. f ∠BAC = 66° and ∠ABC = 80°. Calculate : ∠BIC.

In the given figure, AB = AD = DC = PB and ∠DBC = x°. determine, in terms of x : (i) ∠ABD, (ii) ∠APB

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

In the given figure; ABC, AEQ and CEP are straight lines. Show that ∠APE and ∠CQE are supplementary.

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

If ∠ADC = 32°, find angle BOC

In a cyclic – quadrilateral PQRS, angle PQR = 135°. Sides SP and RQ produced meet at point A : whereas sides PQ and SR produced meet at point B.

If ∠A : ∠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 meets 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. If ∠DCF : ∠F : ∠E = 3 : 5 : 4, Find the angles of the cyclic quadrilateral ABCD.

The following figure shows a circle with PR as its diameter.

If PQ = 7 cm and QR = 3RS = 6 cm, find the perimeter 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 of ∠CAD.

Further, if the length of arc AC is twice the length of arc BC, find : (a) ∠BAC (b) ∠ABC

In cyclic quadrilateral ABCD; AD = BC, ∠BAC = 30° and ∠CBD = 70°; find:

(i) ∠BCD (ii) ∠BCA (iii) ∠ABC (iv) ∠ADC

In the given figure, ∠ACE = 43° and ∠CAF = 62° ; Find the values of a, b and c.

In the given figure, AB is parallel to DC ∠BCE = 80° and ∠BAC = 25°.

Find:

(i) ∠CAD (ii) ∠CBD (iii) ∠ADC

ABCD is a cyclic quadrilateral 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 = 60°

In the figure, given alongside, CP bisects angle ACB. Show that DP bisects angle ADB.

In the figure, given below, AD = BC, ∠BAC = 30° and ∠CBD = 70°.

Find: (i) ∠BCD (ii) ∠BCA (iii) ∠ABC (iv) ∠ADB

In the figure given below, an AD is a diameter. O is the centre of the circle AD is parallel to BC and `angle CBD` = 32°. Find `angle OBD`

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

Find: ∠AOB

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

Find: ∠BED

In the figure given, O is the centre of the circle. `angle "DAE" = 70^@`, Find giving suitable reasons the measure of:

1) `angle "BCD"`

2) `angle "BOD"`

3) `angle "OBD"`

## Chapter 17: Circles

## Selina solutions for Concise Mathematics Class 10 ICSE chapter 17 - Circles

Selina solutions for Concise Mathematics Class 10 ICSE chapter 17 (Circles) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CISCE Concise Mathematics Class 10 ICSE solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Selina textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Concise Mathematics Class 10 ICSE chapter 17 Circles are Tangent to a Circle, Number of Tangents from a Point on a Circle, Chord Properties - a Straight Line Drawn from the Center of a Circle to Bisect a Chord Which is Not a Diameter is at Right Angles to the Chord, Chord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof), Chord Properties - Equal Chords Are Equidistant from the Center, Chord Properties - Chords Equidistant from the Center Are Equal (Without Proof), Chord Properties - There is One and Only One Circle that Passes Through Three Given Points Not in a Straight Line, Arc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle, Arc and Chord Properties - Angles in the Same Segment of a Circle Are Equal (Without Proof), Arc and Chord Properties - Angle in a Semi-circle is a Right Angle, Arc and Chord Properties - If Two Arcs Subtend Equal Angles at the Center, They Are Equal, and Its Converse, Arc and Chord Properties - If Two Chords Are Equal, They Cut off Equal Arcs, and Its Converse (Without Proof), Arc and Chord Properties - If Two Chords Intersect Internally Or Externally Then the Product of the Lengths of the Segments Are Equal, Cyclic Properties, Concept of Circles, Areas of Sector and Segment of a Circle, Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments, Tangent Properties - If a Chord and a Tangent Intersect Externally, Then the Product of the Lengths of Segments of the Chord is Equal to the Square of the Length of the Tangent from the Point of Contact to the Point of Intersection, Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers.

Using Selina Class 10 solutions Circles exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Selina Solutions are important questions that can be asked in the final exam. Maximum students of CISCE Class 10 prefer Selina Textbook Solutions to score more in exam.

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