1   Fill in the blanks$\\$ (i) The centre of a circle lies in __________ of the circle. (exterior/interior)$\\$ (ii) A point, whose distance from the centre of a circle is greater than its radius lies in __________ of the circle. (exterior/interior)$\\$ (iii) The longest chord of a circle is a __________ of the circle.$\\$ (iv) An arc is a __________ when its ends are the ends of a diameter.$\\$ (v) Segment of a circle is the region between an arc and __________ of the circle.$\\$ (vi) A circle divides the plane, on which it lies, in __________ parts.

Solution :

(i) The centre of a circle lies in $\underline{\text{interior}}$ of the circle.$\\$ (ii) A point, whose distance from the centre of a circle is greater than its radius lies in $\underline{\text{exterior}}$ of the circle.$\\$ (iii) The longest chord of a circle is a $\underline{\text{diameter}}$ of the circle.$\\$ (iv) An arc is a $\underline{\text{semi-circle}}$ when its ends are the ends of a diameter.$\\$ (v) Segment of a circle is the region between an arc and $\underline{\text{chord }} $of the circle.$\\$ (vi) A circle divides the plane, on which it lies, in $\underline{\text{three}}$ parts.

2   Write True or False: Give reasons for your answers.$\\$ (i) Line segment joining the centre to any point on the circle is a radius of the circle.$\\$ (ii) A circle has only finite number of equal chords.$\\$ (iii) If a circle is divided into three equal arcs, each is a major arc.$\\$ (iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.$\\$ (v) Sector is the region between the chord and its corresponding arc.$\\$ (vi) A circle is a plane figure.$\\$

Solution :

(i)True.$\\$ All the points on the circle are at equal distances from the centre of the circle, and this equal distance is called as radius of the circle.$\\$ (ii)False$\\$ There are infinite points on a circle. Therefore, we can draw infinite number of chords of given length. Hence, a circle has infinite number of equal chords.$\\$ (iii) False$\\$ Consider three arcs of same length as $AB, BC,$ and $CA$. It can be observed that for minor arc $BDC, CAB$ is a major arc. Therefore, $AB, BC$, and $CA$ are minor arcs of the circle.$\\$ (iv) True.$\\$ Let $AB$ be a chord which is twice as long as its radius. It can be observed that in this situation, our chord will be passing through the centre of the circle.Therefore, it will be the diameter of the circle.$\\$ (v) False.$\\$ Sector is the region between an arc and two radii joining the centre to the end points of the arc. For example, in the given figure, $OAB$ is the sector of the circle.$\\$ (vi) True.$\\$ A circle is a two-dimensional figure and it can also be referred to as a plane figure.

3   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.

Solution :

A circle is a collection of points which are equidistant from a fixed point. This fixed point is called as the centre of the circle and this equal distance is called as radius of the circle. And thus, the shape of a circle depends on its radius. Therefore, it can be observed that if we try to superimpose two circles of equal radius, then both circles will cover each other. Therefore, two circles are congruent if they have equal radius. Consider two congruent circles having centre $O$ and $O'$ and two chords $AB$ and $CD$ of equal lengths.$\\$ In $\Delta AOB $ and $\Delta CO’D,$$\\$ $AB = CD$ (Chords of same length)$\\$ $OA = O'C$ (Radii of congruent circles)$\\$ $OB = O'D $(Radii of congruent circles)$\\$ $\therefore \Delta AOB \cong \Delta CO'D$ (SSS congruence rule)$\\$ $\Rightarrow \angle AOB = \angle CO'D$ (By $CPCT)$$\\$ Hence, equal chords of congruent circles subtend equal angles at their centres.

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

Solution :

Let us consider two congruent circles (circles of same radius) with centres as $O$ and $O'.$ In $\Delta AOB$ and $\Delta CO'D,$$\\$ $\angle AOB = \angle CO'D$ (Given)$\\$ $OA = O'C$ (Radii of congruent circles)$\\$ $OB = O'D$ (Radii of congruent circles)$\\$ $\therefore \Delta AOB \cong \Delta CO'D$ ($SSS$ congruence rule)$\\$ $ AB = CD $(By $CPCT)$$\\$ Hence, if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

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

Solution :

Consider the following pair of circles.$\\$ The above circles do not intersect each other at any point. Therefore, they do not have any point in common.$\\$ The above circles touch each other only at one point $Y$. Therefore, there is $1$ point in common.$\\$ The above circles touch each other at $1$ point $X$ only. Therefore, the circles have $1$ point in common.$\\$ These circles intersect each other at two points $G$ and $H.$ Therefore, the circles have two points in common. It can be observed that there can be a maximum of $2$ points in common. Consider the situation in which two congruent circles are superimposed on each other. This situation can be referred to as if we are drawing the circle two times.