If $AB = 12 \, cm$,$BC = 16 \, cm$ and $AB$ is perpendicular to $BC$,then the radius of the circle passing through the points $A, B$ and $C$ is (in $cm$)

Write True or False and justify your answer in each of the following: Through three collinear points a circle can be drawn.

If $ABC$ is an equilateral triangle inscribed in a circle and $P$ is any point on the minor arc $BC$ which does not coincide with $B$ or $C$,prove that $PA$ is the angle bisector of $\angle BPC$.

$O$ is the circumcentre of the triangle $ABC$ and $D$ is the mid-point of the base $BC$. Prove that $\angle BOD = \angle A$.

State whether each of the following statements is true or false:
$(1)$ $A$ piece of a circle between two points on a circle is called an arc of the circle.
$(2)$ The length of the complete circle is called its circumference.

State whether each of the following statements is true or false:
$(1)$ The collection of all the points in a plane,which are at a fixed distance from a fixed point in the plane,is called a circle.
$(2)$ The line segment joining the centre and any point on the circle is called a radius of the circle.