What measure of the circles make them congruent?

‘O’ is the centre of the circle. The area of the shaded region in the given figure is $126 c{m}^2 , \overline{PQ} = \overline{PR}$ then the diameter $QR$ is:

In the given figure, three circles with centres $A, B, C$ respectively touch each other externally. If $AB=5cm, BC=7cm$ and $CA=6cm$, then the radius of the circle with centre $A$ is

From the figure of $\odot \left(O,r\right)$ match the following. The correct option is _____

Match

$\left(1\right) \overline{AC}\cup \stackrel{⏜}{ABC}$                 $\left(a\right)2r$

$\left(2\right)\overline{OA}$ length                      $\left(b\right)r$

$\left(3\right)\overline{AC}$ length                      $\left(c\right)\mathrm{\pi r}+2\mathrm{r}$

Chords $AB$ and $CD$ are intersecting at $P$. $AB=10$ centimetres, $PB=4$ centimetres and $PD=3$ centimetres. What is the length of $PA$?

What is the circumference of the largest circle which can be inscribed in a square of side $14 cm$? ($\mathrm{\pi }=\frac{22}{7}$)

In the figure, $a, b, c$ are the sides of a right triangle where $c$ is the hypotenuse. Prove that the radius $r$ of the circle which touches the sides of the triangle is given by $r=\frac{a+b+c}{2}$.