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.

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°.

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 of radius 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.

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.

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

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 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.

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

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

A line segment AB is 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.

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

Given an arc of a circle, complete the circle.

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

Suppose you are given a circle. Give a construction to find its centre.

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 centre. If the distance between AB and CD is 6 cm, find the radius of the circle.

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?

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 and Nisha to Ishita. If the distance between Ishita and Isha and between Isha and Nisha is 24 m each, what is 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.

In Fig. 16.120, O is the centre of the circle. If ∠APB=50°, find ∠AOB and ∠OAB.

In Fig. 16.121, it is given that O is the centre of the circle and ∠AOC=150°. Find ∠ABC.

In Fig. 16.122, O is the centre of the circle. Find ∠BAC.

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

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

In Fig. 16.135, O is the centre of the circle, Bo is the bisector of ∠ABC. Show that AB=AC.

In Fig. 16. 136, O is the centre of the circle, prove that ∠x=∠y+∠z.

In Fig. 16.137, O and O’ are centres of two circles intersecting at B and C. ACD is a straight line, find x.

In Fig. 16.138, O is the centre of a circle and PQ is a diameter. If ∠ROS=40°, find ∠RTS.

In Fig. 16.139, if ∠ACB=40°, ∠DPB=120°, find ∠CBD.

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

In Fig. 16.176, Δ ABC is an equilateral triangle. Find m∠BEC.

In Fig. 16.177, Δ PQR is an isosceles triangle with PQ=PR and m∠PQR=35°. Find m∠QSR and m∠QTR.

In Fig. 16.178, O is the centre of the circle. If ∠BOD=160°, find the values of x and y.

In Fig. 16.179 ABCD is a cyclic quadrilateral. If ∠BCD=100° and ∠ABD=70°, find ∠ADB.

If ABCD is a cyclic quadrilateral in which AD||BC (fig. 16.180). Prove that ∠B=∠C.

In Fig. 16.181, O is the centre of the circle. Find ∠CBD.

In Fig. 16.182, 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 a cyclic quadrilateral ABCD if AB||CD and B=70°, find the remaining angles.

In a cyclic quadrilateral ABCD, if m ∠A =3(m∠C). Find m∠A.

In Fig. 16.183, O is the centre of the circle ∠DAB=50°. Calculate the values of x and y.

In Fig. 16.184, if ∠BAC=60° and ∠BCA=20°, find ∠ADC.

In Fig. 16.185, if ABC is an equilateral triangle. Find ∠BDC and ∠BEC

In Fig. 16.186, O is the centre of the circle. If ∠CEA=30°, find the values of x, y and z.

In Fig. 16.187, ∠BAD=78°, ∠DCF=x° and ∠DEF=y°. Find the values of x and y.

In a cyclic quadrilateral ABCD, if ∠A-∠C=60°, prove that the smaller of two is 60°.

In Fig. 16.188, ABCD is a cyclic quadrilateral. Find the value of x.

ABCD is a cyclic quadrilateral in which:

(i) BC||AD, ∠ADC=110° and ∠BAC=50°. Find ∠DAC.

(ii) ∠DBC=80° and ∠BAC=40° Find ∠BCD.

(iii) ∠BCD=100° and ∠ABD=70°. Find ∠ADB.

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.

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

If the two sides of a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are equal.

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

(i) AD||BC (ii) EB=EC

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).

Prove that the angle in a segment shorter than a semicircle is greater than a right angle.

Prove that the angle in a segment greater than a semi-circle is less than a right angle.

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

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.

In Fig. 16.189, ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If ∠DBC=55° and ∠BAC=45°, find ∠BCD.

In Fig. 16.193, two circles intersect at A and B. The centre of the smaller circle is O and it lies on the circumference of the larger circle. If ∠APB=70°, find ∠ACB.

In Fig. 16.194, two congruent circles with centres O and O’ intersect at A and B. If ∠AO’B=50°, then find ∠APB.

In Fig. 16.195, ABCD is a cyclic quadrilateral in which ∠BAD=75°, ∠ABD=58° and ∠ADC=77°, AC and BD intersect at P. Then, find ∠DPC.

In Fig. 16.196, if ∠AOB = 80° and ∠ABC=30°, then find ∠CAO.

In Fig. 16.196, if O is the circumcentre of Δ ABC, then find the value of ∠OBC + ∠BAC.

In Fig. 16.197, AOC is a diameter if the circle and arc AXB=arc BYC. Find ∠BOC.

In Fig. 16.198, A is the centre of the circle. ABCD is a parallelogram and CDE is a straight line. Find ∠BCD:∠ABE

In Fig. 16.199, AB is a diameter of the circle such that ∠A=35° and ∠Q=25°, find ∠PBR.

In Fig. 16.200, P and Q are centres of two circles intersecting at B and C. ACD is a straight line. Then, ∠BQD=

In Fig. 16.201, 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 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 A as 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 a centre and AD diameter, than ∠AOB =

Let C be the mid-point of an arc AB of a circle such that m=183°. If the region bounded by the arc ACB and 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=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 ∠BOC 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 circles 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

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 to the circumference is

The chord of a circle is equal to its radius. The angle substended 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=

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 in 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

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 Fig. 16.202, O is the centre of the circle such that ∠AOC=130°, then ∠ABC=

In Fig. 16.203, if chords AB and CD of the circle intersect each other at right angles, then x + y =

In Fig. 16.204, if ∠ABC= 45°, then ∠AOC=

In Fig. 16.205, chords AD and BC intersect each other at right angles at a point P. If ∠DAB=35°, then ∠ADC=

In Fig. 16.206, O is the centre of the circle and ∠BDC=42°. The measure of ∠ACB is