In the given figure, the perpendicular bisector $\mathrm{AD}$ of the equilateral triangle $\mathrm{ABC}$ is the same as the diameter of the circle $\mathrm{O},$ and the circle intersects the sides $\mathrm{AB}$ and $\mathrm{AC}$ at $\mathrm{P}$ and $\mathrm{Q},$ respectively. If the height of the triangle be, $12\sqrt{2} \mathrm{cm},$ find the total area of the shaded region.

There are two concentric circles centered at $\mathrm{O}.$ Their radii are $21 \mathrm{cm}$ and $29 \mathrm{cm}. \mathrm{AB}$ is the diameter of the larger circle and $\mathrm{BC}$ is the tangent to the smaller circle at $\mathrm{C}.$ Find $\mathrm{AC}.$

Let there be a circle with centre $\mathrm{O}$ and radius $\mathrm{r},$ such that the four points. $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$ are cyclic while $\mathrm{AC}$ and $\mathrm{BD}$ are the two chords intersecting each other perpendicularly at a point $\mathrm{P}$ somewhere inside the circle. If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$ are variable points and $\mathrm{OP}=\mathrm{d},$ find the maximum and minimum area of the quadrilateral $\mathrm{ABCD}.$

Two equal circles with centers $\mathrm{P}$ and $\mathrm{Q}$ are tangent at $\mathrm{O} . \mathrm{A}$ common line is tangent to both the circles at $\mathrm{M}$ and $\mathrm{N}$ respectively. Point $\mathrm{B}$ lies on the arc $\mathrm{OM}$ and point $\mathrm{C}$ lies on the arc $\mathrm{ON}.$ Points $\mathrm{A}$ and $\mathrm{D}$ lie on the tangent $\mathrm{MN}.$ If the radius of each circle is $\mathrm{r}$ and each side of the square $\mathrm{ABCD}$ is $\mathrm{x},$ find $\mathrm{x}.$

In the following diagram four circles are tangent to every other circle. The radius of each of the circles $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ is $1.$ Find the radius of circle $\mathrm{D},$ which is tangent to all the three given circles and lies at the centre.

In the following diagram there are three circles packed in another circle, such that each circle is tangent to all other circles in this arrangement. The radius of the circumscribing circle $\mathrm{A}$ is $1 \mathrm{cm};$ and circle $\mathrm{B}$ and circle $\mathrm{C}$ are congruent. Find the radius of the circle $\mathrm{D}.$

In the following diagram there are four circles $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$ packed in a circle $\mathrm{P}.$ The radius of the circumscrib $\mathrm{g}$ circle $\mathrm{P}$ is $1 \mathrm{cm}.$ Circles $\mathrm{A}$ anc $\mathrm{B}$ are tangent at $\mathrm{P},$ the centre of the circumscribing circle. Circle $\mathrm{D}$ is tangent to circles $\mathrm{A}, \mathrm{C}$ and $\mathrm{P}.$ Find the radius of the circle $\mathrm{D}.$