If the radii of two concentric circles are $4 \, cm$ and $5 \, cm$,then the length of each chord of the larger circle which is tangent to the smaller circle is (in $cm$):

$\overline{PA}$ and $\overline{PB}$ are the tangents to $\odot(O, r)$ drawn from a point $P$ outside a circle. If $m \angle APB = 70^\circ$,then $m \angle POB = \dots$ (in $^\circ$)

If $d_{1}$ and $d_{2}$ $(d_{2} > d_{1})$ are the diameters of two concentric circles and $c$ is the length of a chord of the outer circle which is tangent to the inner circle,prove that $d_{2}^{2} = c^{2} + d_{1}^{2}$.

Write 'True' or 'False' and give reasons for your answer.
$AB$ is a diameter of a circle and $AC$ is its chord such that $\angle BAC = 30^{\circ}$. If the tangent at $C$ intersects $AB$ extended at $D$,then $BC = BD$.

$P$ is in the exterior of $\odot (O, 9)$. $A$ tangent from $P$ touches the circle at $T$. If $PT = 40$,then $OP = \ldots$

If $a, b, c$ are the sides of a right triangle where $c$ is the hypotenuse,prove that the radius $r$ of the incircle of the triangle is given by $r = \frac{a + b - c}{2}$.