RD Sharma solutions for Mathematics [English] Class 9 chapter 15 - Circles [Latest edition]

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All points lying inside/outside a circle are called .................. points/ .....................points.

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Circles having the same centre and different radii are called ...........................circles.

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A point whose distance from the centre of a circle is greater than its radius lies in ..................... of the circle.

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A continuous piece of a circle is ............... of the circle

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The longest chord of a circle is a ................of the circle.

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An arc is a ................ when its ends are the ends of a diameter.

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Segment of a circle is the region between an arc and .................. of the circle.

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A circle divides the plane, on which it lies, in ............ parts.

true or false 

A circle is a plane figure.

true or false 

Line segment joining the centre to any point on the circle is a radius of the circle,

true or false 

If a circle is divided into three equal arcs each is a major arc.

true or false

A circle has only finite number of equal chords.

true or false 

A chord of a circle, which is twice as long is its radius is a diameter of the circle.

true or false

Sector is the region between the chord and its corresponding arc.

true or false 

The degree measure of an arc is the complement of the central angle containing the arc.

ture or false v

The degree measure of a semi-circle is 180°.

The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.

Find the length of a chord which is at a distance of 5 cm from the centre of a circle ofradius 10 cm.

Find the length of a chord which is at a distance of 4 cm from the centre of the circle of radius 6 cm.

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

A line segment AB is of length 5 cm. Draw a circle of radius 4 cm passing through A and B. Can you draw a circle of radius 2 cm passing through A and B? Give reason in support of your answer.

Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.

An equilateral triangle of side 9cm is inscribed in a circle. Find the radius of the circle.

Given an arc of a circle, show how to complete the circle.

Given an arc of a circle, complete the circle.

Suppose You Are Given a Circle. Give a Construction to Find Its Centre.

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

Two chords AB, CD of lengths 5 cm, 11 cm respectively of a circle are parallel. If the distance between AB and CD is 3 cm, find the radius of the circle.

Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

Prove that the line joining the mid-point of a chord to the centre of the circle passes through the mid-point of the corresponding minor arc.

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

Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are opposite side of its center. If the distance between AB and CD is 6 cm. Find the radius of the circle.

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 o Isha, Isha to Nisha and 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.

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

In the below 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

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

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 figures.

If O is the centre of the circle, find the value of x in the following figures.

O is the circumcentre of the triangle ABC and OD is perpendicular on BC. Prove that ∠BOD = ∠A

In the given figure, O is the centre of the circle, BO is the bisector of ∠ABC. Show that AB = BC. 

In the given figure, O and O' are centres of two circles intersecting at B and C. ACD is a straight line, find x. 

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

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

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

In the given figure, ΔABC is an equilateral triangle. Find m∠BEC. 

In the given figure, ΔPQR is an isosceles triangle with PQ = PR and m ∠PQR = 35°. Find m ∠QSR and m ∠QTR.

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

In the given figure, ABCD is a cyclic quadrilateral. If ∠BCD = 100° and ∠ABD = 70°, find ∠ADB.

If ABCD is a cyclic quadrilateral in which AD || BC (In the given figure). Prove that ∠B = ∠C.

In the given figure, O is the centre of the circle. Find ∠CBD.

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

On a semi-circle with AB as diameter, a point C is taken, so that m (∠CAB) = 30°. Find m(∠ACB) and m (∠ABC).

In the given figure, O is the centre of the circle and ∠DAB = 50° . Calculate the values of xand y. 

In the given figure, if ∠BAC = 60° and ∠BCA = 20°, find ∠ADC. 

In the given figure, if ABC is an equilateral triangle. Find ∠BDC and ∠BEC.

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

In the given figure, ∠BAD = 78°, ∠DCF = x° and ∠DEF = y°. Find the values of x and y. 

In the given figure, ABCD is a cyclic quadrilateral. Find the value of x.

ABCD is a cyclic quadrilateral in  BC || AD, ∠ADC = 110° and ∠BAC = 50°. Find ∠DAC.

ABCD is a cyclic quadrilateral in  ∠DBC = 80° and ∠BAC = 40°. Find ∠BCD.

ABCD is a cyclic quadrilateral in   ∠BCD = 100° and ∠ABD = 70° find ∠ADB.

Prove that the circles described on the four sides of a rhombus as diameters, pass through the point of intersection of its diagonals. 

Circles are described on the sides of a triangle as diameters. Prove that the circles on any two sides intersect each other on the third side (or third side produced).

ABCD is a cyclic trapezium with AD || BC. If ∠B = 70°, determine other three angles of the trapezium.

In the given figure, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If ∠DBC = 55° and ∠BAC = 45°, find ∠BCD.

Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.

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

ABCD is a cyclic quadrilateral in which BA and CD when produced meet in E and EA = ED. Prove that AD || BC . 

ABCD is a cyclic quadrilateral in which BA and CD when produced meet in E and EA = ED. Prove that  EB = EC. 

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

In the given figure, two circles intersect at A and B. The centre of the smaller circle is Oand it lies on the circumference of the larger circle. If ∠APB = 70°, find ∠ACB.

In the given figure, two congruent circles with centres O and O' intersect at A and B. If ∠AOB = 50°, then find ∠APB.

In the given figure, ABCD is a cyclic quadrilateral in which ∠BAD = 75°, ∠ABD = 58° and ∠ADC = 77°, AC and BD intersect at P. Then, find ∠DPC.

In the given figure, if ∠AOB = 80° and ∠ABC = 30°, then find ∠CAO. 

In the given figure, A is the centre of the circle. ABCD is a parallelogram and CDE is a straight line. Find ∠BCD : ∠ABE.

In the given figure, AB is a diameter of the circle such that ∠A = 35° and ∠Q = 25°, find ∠PBR.

In the given figure, P and Q are centres of two circles intersecting at B and C. ACD is a straight line. Then, ∠BQD =

In the given figure, if O is the circumcentre of ∠ABC, then find the value of ∠OBC + ∠BAC.

If the given figure, AOC is a diameter of the circle and arc AXB = \[\frac{1}{2}\] arc BYC. Find ∠BOC.

In the given figure, ABCD is a quadrilateral inscribed in a circle with centre O. CD is produced to E such that ∠AED = 95° and ∠OBA = 30°. Find ∠OAC.

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

The radius of a circle is 6 cm. The perpendicular distance from the centre of the circle to the chord which is 8 cm in length, is

If O is the centre of a circle of radius r and AB is a chord of the circle at a distance r/2 from O, then ∠BAO =

ABCD is a cyclic quadrilateral such that ∠ADB = 30° and ∠DCA = 80°, then ∠DAB =

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

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

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

If AB, BC and CD are equal chords of a circle with O as centre and AD diameter, than ∠AOB =

Let C be the mid-point of an arc AB of a circle such that m \[ \stackrel\frown{AB}\]  = 183°. If the region bounded by the arc ACB and the line segment AB is denoted by S, then the centre O of the circle lies

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

If A and B are two points on a circle such that m \[ \stackrel\frown{AB}\] = 260°. A possible value for the angle subtended by arc BA at a point on the circle is 

An equilateral triangle ABC is inscribed in a circle with centre O. The measures of ∠BOCis

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

If ABC is an arc of a circle and ∠ABC = 135°, then the ratio of arc \[\stackrel\frown{ABC}\] to the circumference is ______.

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

PQRS is a cyclic quadrilateral such that PR is a diameter of the circle. If ∠QPR = 67° and ∠SPR = 72°, then ∠QRS =

If A , B, C are three points on a circle with centre O such that ∠AOB = 90° and ∠BOC = 120°, then ∠ABC =

The greatest chord of a circle is called its

Angle formed in minor segment of a circle is

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

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

In the given figure, if ∠ABC = 45°, then ∠AOC =

In the given figure, chords AD and BC intersect each other at right angles at a point P. If ∠DAB = 35°, then 

In the given figure, O is the centre of the circle and ∠BDC = 42°. The measure of ∠ACB is

In a circle with centre O, AB and CD are two diameters perpendicular to each other. The length of chord AC is

Two equal circles of radius r intersect such that each passes through the centre of the other. The length of the common chord of the circle is

If AB is a chord of a circle, P and Q are the two points on the circle different from A and B, then

AB and CD are two parallel chords of a circle with centre O such that AB = 6 cm and CD= 12 cm. The chords are on the same side of the centre and the distance between them is 3 cm. The radius of the circle is

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

In the given figure, O is the centre of the circle such that ∠AOC = 130°, then ∠ABC =