**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.

**Answer**

(i) True.

All the line segment from the centre to the circle is of equal length.

(ii) False.

We can draw infinite numbers of equal chords.

(iii) False.

We get major and minor arcs for unequal arcs. So, for equal arcs on circle we can't say it is major arc or minor arc.

(iv) True.

A chord which is twice as long as radius must pass through the centre of the circle and is diameter to the circle.

(v) False.

Sector is the region between the arc and the two radii of the circle.

(vi) True.

A circle can be drawn on the plane.

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.

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 ΔAOB and ΔCO'D,

AB = CD (Chords of same length)

OA = O'C (Radii of congruent circles)

OB = O'D (Radii of congruent circles)

∴ ΔAOB ≅ ΔCO'D (SSS congruence rule)

⇒ ∠ AOB = ∠ CO'D (By CPCT)

Hence, equal chords of congruent circles subtend equal angles at their centres

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

Let us consider two congruent circles (circles of same radius) with centres as O and O'.

In ΔAOB and ΔCO'D,

∠ AOB = ∠ CO'D (Given)

OA = O'C (Radii of congruent circles)

OB = O'D (Radii of congruent circles)

∴ ΔAOB ≅ ΔCO'D (SAS congruence rule)

⇒ AB = CD (By CPCT)

Hence, if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

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

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.