$A$ circle touches the sides $\overline{AB}$,$\overline{BC}$,and $\overline{CA}$ of $\Delta ABC$ at the points $D, E, F$ respectively. If $AB=13$,$BC=12$,and $CA=5$,then $AD = \ldots$

$A$ circle touches all the sides of a quadrilateral $ABCD$. Then, the quadrilateral $ABCD$ is a:

Write 'True' or 'False' and give reasons for your answer.
If a chord $AB$ subtends an angle of $60^{\circ}$ at the centre of a circle,then the angle between the tangents at $A$ and $B$ is also $60^{\circ}$.

$A$ line drawn from the centre $O$ of a circle with radius $7 \, cm$ intersects the tangent at point $P$ on the circle at point $Q$,such that $PQ = 24 \, cm$. Find $OQ$.

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

Which of the following groups correctly matches the data of Part $I$ with the data of Part $II$?

Part $I$	Part $II$
$1.$ In $\Delta ABC, AB=3, BC=4, AC=5$	$a.$ In-radius $= 1$
$2.$ In $\Delta PQR, PQ=5, QR=12, PR=13$	$b.$ In-radius $= 2$
$3.$ In $\Delta XYZ, XY=8, YZ=15, XZ=17$	$c.$ In-radius $= 3$
$4.$ In $\Delta MNP, MN=20, NP=21, MP=29$	$d.$ In-radius $= 6$