Two circles with centres $O$ and $O^{\prime}$ of radii $3\, cm$ and $4\, cm$,respectively,intersect at two points $P$ and $Q$ such that $OP$ and $O^{\prime}P$ are tangents to the two circles. Find the length of the common chord $PQ$ (in $cm$).

Prove that a diameter $AB$ of a circle bisects all those chords which are parallel to the tangent at the point $A$.

Write 'True' or 'False' and give reasons for your answer.
In the figure,$PQL$ and $PRM$ are tangents to the circle with center $O$ at the points $Q$ and $R$ respectively,and $S$ is a point on the circle such that $\angle SQL = 50^{\circ}$ and $\angle SRM = 60^{\circ}$. Then $\angle QSR$ is equal to $40^{\circ}$.

In $\Delta ABC$,$\angle B$ is a right angle. If $AB = 8$ and $BC = 15$,find the radius of the incircle of $\Delta ABC$.

At one end $A$ of a diameter $AB$ of a circle of radius $5 \, cm$,a tangent $XY$ is drawn to the circle. The length of the chord $CD$ parallel to $XY$ and at a distance of $8 \, cm$ from $A$ is (in $cm$):

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.