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

Two circles intersect at two points $B$ and $C$. Through $B$,two line segments $ABD$ and $PBQ$ are drawn to intersect the circles at $A, D$ and $P, Q$ respectively (see figure). Prove that $\angle ACP = \angle QCD$.

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

State whether the following statements are True or False. Give reasons for your answers.
$(i)$ The line segment joining the centre to any point on the circle is a radius of the circle.
$(ii)$ $A$ circle has only a 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.
$(v)$ $A$ sector is the region between the chord and its corresponding arc.
$(vi)$ $A$ circle is a plane figure.

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.

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.