# RD SHARMA Solutions for Class 9 Maths Chapter 15 - Circles

## Chapter 15 - Circles Exercise Ex. 15.1

**Fill in the blanks:**** (i) All points lying inside/outside a circle are called ...... points/ ... points.**** (ii) Circles having the same centre and different radii are called ... circles.**** (iii) A point whose distance from the centre of a circle is greater than its radius lies in ... of the circle.**** (iv) A continuous piece of a circle is ... of the circle.**** (v) The longest chord of a circle is a ... of the circle. **** (vi) An arc is a ... when its ends are the ends of a diameter. **** (vii) Segment of a circle is the region between an arc and ... of the circle. **** (viii) A circle divides the plane, on which it lies, in .... parts.**

**(i) interior/exterior**** (ii) concentric**** (iii) the exterior**** (iv) arc**** (v) diameter**** (vi) semi-circle**** (vii) centre**** (viii) three**

**Write the truth value (T/F) of the following with suitable reasons:**** (i) A circle is a plane figure. **** (ii) Line segment joining the centre to any point on the circle is a radius of the circle. **** (iii) If a circle is divided into three equal arcs each is a major arc. **** (iv) A circle has only finite number of equal chords. **** (v) A chord of a circle, which is twice as long is its radius is a diameter of the circle. **** (vi) Sector is the region between the chord and its corresponding arc. **** (vii) The degree measure of an arc is the complement of the central angle containing the arc. **** (viii) The degree measure of a semi-circle is 180 ^{o}.**

**(i) T**** (ii) T**** (iii) T**** (iv) F**** (v) T**** (vi) T**** (vii) F**** (viii) T**

## Chapter 15 - Circles Exercise Ex. 15.2

Give a method to find the centre of a given circle.

Steps of construction:

(1) Take three point A, B and C on the given circle.

(2) Join AB and BC.

(3) Draw the perpendicular bisectors of chord AB and BC which interesect each other at O.

(4) Point O will be the required circle because we know that the perpendicular bisector of a chord always passes through the centre.

The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord form the centre?

OM = 4 cm

Prove that two different circles cannot intersect each other at more than two points.

Suppose two different circles can intersect each other at three points then they will pass through the three common points but we know that there is one and only one circle with passes through three non-collinear points, which contradicts our supposition.

Hence, two different circles cannot intersect each other at more than two points.

In MOB

So, from equation (1) and (2)

## Chapter 15 - Circles Exercise Ex. 15.3

Three girls Ishita, Isha and Nisha are playing a game by standing on a circle of radius 20 m drawn in a park. Ishita throws a ball to Isha, Isha to Nisha, Nisha to Ishita. If the distance between Ishita and Isha and between Isha and Nisha is 24 m each, what is the distance between Ishita and Nisha?

A circular park of radius 40 m is situated in a colony. Three boys Ankur, Amit and Anand are sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.

## Chapter 15 - Circles Exercise Ex. 15.4

In fig., O is the centre of the circle. If ∠APB = 50°, find ∠AOB and ∠OAB.

In fig., O is the centre of the circle. Find ∠BAC.

**If O is the centre of the circle. Find the value of x in the following figure:**

**If O is the centre of the circle. Find the value of x in the following figure:**

**If O is the centre of the circle. Find the value of x in the following figure:**

**If O is the centre of the circle. Find the value of x in the following figure:**

In fig., O is the centre of the circle, BO is the bisector of ∠ABC. Show that AB = AC.

**In fig., O and O' are centres of two circles intersecting at B and C. ABD is straight line, find x.**

In fig., if ∠ACB = 40°, ∠DPB = 120°, find ∠CBD.

**A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.**

In fig., it is given given that O is the centre of the circle and ∠AOC = 150°. Find ∠ABC.

In fig., O is the centre of the circle, prove that ∠x = ∠y + ∠z.

in fig., O is the centre of a circle and PQ is a diameter. If ∠ROS = 40°, find. ∠RTS.

## Chapter 15 - Circles Exercise Ex. 15.5

In fig., ΔABC is an equilateral triangle. Find m∠BEC.

In fig., ΔPQR is an isosceles triangle with PQ = PR and m∠PQR = 35°. find m∠QSR and m∠QTR.

In fig., O is the centre of the circle. If ∠BOD = 160°, find the values of x and y.

In fig., ABCD is a cyclic qudrilateral. If ∠BCD = 100° and ABD = 70°, find ∠ADB.

If ABCD is a cyclic quadrilateral in which AD ∥ BC. Prove that ∠B = ∠C.

In fig., O is the centre of the circle. find ∠CBD.

In fig., AB and CD are diameters of a circle with centre O. If ∠OBD = 50°, find ∠AOC.

In fig., O is the centre of the circle and DAB = 50. calculate the values of x and y.

In fig., if ∠BAC = 60°, and ∠BCA = 20°, find ∠ADC.

In fig., if ABC is an equilateral triangle. Find ∠BDC and ∠BEC.

In fig., O is the centre of the circle. If ∠CEA = 30°, find the values of x, y and z.

In fig., ∠BAD = 78°, ∠DCF = x° and DEF = y° find the values of x and y.

In fig., ABCD is cyclic qudrilateral. Find the value of x.

In fig., ABCD is cyclic quadrilaterial in which AC an BD are its diagonals. If ∠DBC = 55° and ∠BAC = 45°, find ∠BCD.

**Prove that the centre of the circle circumscribing the cyclic rectangle ABCD is the point of intersection of its diagonals.**

**Let O be the centre of the circle circumscribing the cyclic rectangle ABCD. Since ABC = 90 ^{o} and AC is a chord of the circle, so, AC is a diameter of the circle. Similarly, BD is a diameter. Hence, point of intersection of AC and BD is the centre of the circle. **

**Prove that the line segment joining the mid-point of the hypotenuse of a right triangle to its opposite vertex is half of the hypotenuse.**

## Chapter 15 - Circles Exercise 15.109

If the length of a chord of a circle is 16 cm and is at a distance of 15 cm from the centre of the circle, then the radius of the circle is

(a) 15 cm

(b) 16 cm

(c) 17 an

(d) 34 cm

## Chapter 15 - Circles Exercise 15.110

(a) 60°

(b) 45°

(c) 30°

(d) 15°

A chord of length 14 cm is at a distance of 6 cm from the centre of a circle. The length of another chord at a distance of 2 cm from the centre of the circle is

(a) 12 cm

(b)) 14 cm

(c) 16 cm

(d) 18 cm

One chord of a circle is known to be 10 cm. The radius of this circle must be

(a) 5 cm

(b) greater than 5 cm

(c) greater than or equal to 5 cm

(d) less than 5 cm

ABC is a triangle with B as right angle, AC = 5 cm and AB = 4 cm. A circle is drawn with A as centre and AC as radius. The length of the chord of this circle passing through C and B is

(a) 3 cm

(b) 4 cm

(c) 5 cm

(d) 6 cm

In a circle, the major arc is 3 times the minor arc. The corresponding central angles and the degree measures of two arcs are

(a) 90° and 270°

(b) 90° and 90°

(c) 270° and 90°

(d) 60° and 210°

If two diameters of a circle intersect each other at right angles, then quadrilateral formed joining their end points is a

(a) rhombus

(b) rectangle

(c) parallelogram

(d) square

The chord of a circle is equal to its radius. The angle subtended by this chord at the minor of the circle is

(a) 60^{0 }

(b) 75^{0}

(c) 120^{0}

(d) 150^{0}

## Chapter 15 - Circles Exercise 15.111

The greatest chord of a circle is called its

(a) radius

(b) secant

(c) diameter

(d) none of these

The greatest chord of the circle is diameter of the circle.

Hence, correct option is (c).

Angle formed in minor segment of a circle is

(a) acute

(b) obtuse

(c) right angle

(d) none of these

Angle formed in a minor segment is always a obtuse angle.

Hence, correct option is (b).

Number of circles that can be drawn through three non-collinear points is

(a) 1

(b) 0

(c) 2

(d) 3

Three non-collinear points make a triangle and there is only one circle that can pass through all three points,

i.e. circumcircle of that triangle.

Hence, correct option is (a).

In figure, if chords AB and CD of the circle intersect each other at right angles, then x + y =

(a) 45^{0 }

(b) 60^{0}

(c) 75^{0 }

(d) 90^{0}

In figure, chords AD and BC intersect each other at right angles at a point P. If

(a) 35^{0}

(b) 45^{0}

(c) 55^{0}

(d) 65^{0}

## Chapter 15 - Circles Exercise 15.112

In a circle of radius 17 cm, two parallel chords are drawn on opposite side of a diameter. The distance between the chords is 23 cm. If the length of one chord is 16 cm, then the length of the other is

(a) 34 cm

(b) 15 cm

(c) 23 cm

(d) 30 cm

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