Q. Fill in the blanks:
(1) The centre of a circle lies in ___________ of the circle (exterior/interior)
(2) A point whose distance from the centre of a circle is greater than its radius lies in ___________ of the circle
(3) The longest chord of a circle is a ____________ of the circle.
(4) An arc is a ____________ when its ends are the ends of a diameter.
(5) Segment of a circle is the region between an arc and __________ of the circle.
(6) A circle divides the plane, on which it lies in ______________ parts.

Q. Write true or false: Give reasons for your answers.
(1) Line segment joining the centre to any point on the circle is a radius of the circle.
(2) A circle has only finite number of equal chords.
(3) If a circle is divided into three equal arcs, each is a major arc.
(4) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.
(5) Sector is the region between the chord and its corresponding arc.
(6) A circle is a plane figure.

Q. Prove that if chords of congruent circles subtend equal angles their centres, then the chords are equal.

Q. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

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

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

Q. If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.

Q. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of other chord.

Q. If a line intersects two concentric circles (circles with the same centre) with centre $O$ at $A,B,C$ and $D$, prove that $AB=CD$ (see figure)

Q. A circular park of radius $20m$ is situated in a colony. Three boys Ankur, Syed and David 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.

Q. Two circles of radii $5cm$ and $3cm$ intersect at two points and the distance between their centres is $4cm$. Find the length of the common chord.

Q. Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius $5m$ drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is $6m$ each, what is the distance between Reshma and Mandip?

Q. It two equal chords of a circle intersect within the circle. Prove that the line joining the point of intersection to the centre makes equal angles with the chords.

Q. In the figure, $A,B,C$ and $D$ are four points on a circle. $AC$ and $BD$ intersect at a point $E$, such that $\angle BEC={130}^o$ and $\angle ECD={20}^o$. Find $\angle BAC$.

Q. 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.

Q. $ABCD$ is a cyclic quadrilateral whose diagonals intersect at a point $E$. If $\Delta DBC={70}^o$, $\Delta BAC$ is ${30}^o$, find $\angle BCD$. Further if $AB=BC$, find $\angle ECD$.

Q. In the figure, $\angle PQR={100}^o$, where $P,Q$ and $R$ are points on a circle with centre $O$. Find $\angle OPR$.

Q. In the figure $A,B$ and $C$ are three points on a circle with centre $O$ such that $\angle BOC={30}^o$ and $\angle AOB={60}^o$. If $D$ is a point on the circle other than the arc $ABC$, find $\angle ADC$.

Q. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Q. In the figure, $\angle ABC={69}^o,\angle ACB={31}^o$, find $\angle BDC$.