Which of the following statements is/are correct?
(i) A kite always has two axes of symmetry
(ii) A square is both a kite and an isosceles trapezium
(iii) Every kite is an orthodiagonal
(iv) An isosceles trapezium can have a circumcircle
(v) A kite can always have an incircle
(vi) If the sum of two opposite sides is not the same as the sum of the other pair of opposite sides, it can never be a circumscribing quadrilateral.

Which of the following is correct, considering the given figure, in which $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ are the mid-points of $\mathrm{AB}, \mathrm{BC}, \mathrm{CD}$ and $\mathrm{AD}$, respectively?

(i) The perimeter of the parallelogram $\mathrm{PQRS}$ is equal to the sum of the diagonals of the quadrilateral $\mathrm{ABCD}$
(ii) ${\mathrm{X}}_1+{\mathrm{X}}_3={\mathrm{X}}_2+{\mathrm{X}}_4$
(iii) $\mathrm{A}\left(\mathrm{PQRS}\right)=\frac{1}{2}\mathrm{A}\left(\mathrm{ABCD}\right)$

In the following quadrilateral $\mathrm{ABCD}, \mathrm{E}$ and $\mathrm{F}$ are the mid-points of $\mathrm{DC}$ and $\mathrm{AB}$, respectively. If the area of $∆\mathrm{ADF}$ is $9{ \mathrm{cm}}^2$ and that of $∆\mathrm{EBC}$ is $18{ \mathrm{cm}}^2$, what is the area of quadrilateral $\mathrm{ABCD}$ ?

A trapezium $\mathrm{ABCD}$ inscribes a circle. The two sides $\mathrm{AB}$ and $\mathrm{CD}$ are parallel and the diameter of the circle is equal to the length of side $\mathrm{AD} .$ If $\mathrm{DC}=5 \mathrm{cm}$, find the perimeter of the trapezium $\mathrm{ABCD}$.

In the following quadrilateral $\mathrm{AC}$ and $\mathrm{BD}$ are the diagonals intersecting at a point $\mathrm{O}$ inside the quadrilateral. The area of $∆\mathrm{AOD}$ is $32{ \mathrm{cm}}^2$ and area of $∆\mathrm{BOC}$ is $18{ \mathrm{cm}}^2$. What is the minimum possible area of the quadrilateral $\mathrm{ABCD} ?$

In the following figure if $\mathrm{ABCDE}$ is a regular pentagon, then $\mathrm{ABCD}$ is necessarily

In the following figure if $\mathrm{ABCDE}$ is a regular pentagon, then $\mathrm{AEDF}$ is always

In the following figure, $\mathrm{ABCDE}$ is a regular pentagon. If $\mathrm{AB}=2 \mathrm{cm}$, find the perimeter of the quadrilateral $\mathrm{APDE}$.