Two concentric circles are of radii $3 \mathrm{cm}$ and $5 \mathrm{cm}$. A chord of the greater circle is a tangent to the smaller circle. Find the length of the chord. Write the answer in $\mathrm{cm}$.

The distance between the centre of two circles of radii $3 \mathrm{cm}$ and $8 \mathrm{cm}$ is $13 \mathrm{cm}$. If the points of contact of the direct common tangent of the two circles be $P$ and $Q$, then find the length of $PQ$. Write the answer in $\mathrm{cm}$.

$AB$ is a chord of a circle with centre $O$. The tangent at $B$ meets $AO$ produced at $T$. If $\angle BTA=60°$, find the value of $\angle ABT$.

$AB$ is a chord of a circle with centre $O$ and $AP$ is a tangent. Prove that $\angle AOB=2\angle PAB$.

$AB$ and $AC$ are tangents to a circle with centre $O$. Prove that $AO$ is the perpendicular bisector of the chord $BC$ through the point of contact.

In two concentric circles, two chords $AB$ and $AC$ of the greater circle touches the other circle at $P$ and $Q$. Prove that $PQ\parallel BC$.

If three equal circles touch each other, then prove that their centres are the vertices of an equilateral triangle.

$PQ$ is a diameter of a circle and $A$ is a point on the circumference. $PR$ is perpendicular to the tangent at $A$. Prove that $PA$ bisects $\angle QPR$.

$AB$ is the diameter of a circle. The tangent at $C$ (on the circle) meets $AB$ produced at $D$. Prove that $\angle BDC+2\angle BCD=1$ right angle.