Each side of a square $\mathrm{ABCD}$ is divided in the ratio $1: 4,$ as shown in the following diagram. What is the ratio of area of square $\mathrm{JKLM}$ to that of square$\mathrm{ABCD} ?$

In the following figure, each side of the square is divided in the ratio $2: 3,$ such that $\frac{\mathrm{DG}}{\mathrm{GA}}=\frac{\mathrm{AH}}{\mathrm{HB}}=\frac{\mathrm{BE}}{\mathrm{EC}}=\frac{\mathrm{CF}}{\mathrm{FD}}=\frac{2}{3}.$ Find the ratio of the area of square $\mathrm{PQRS}$ to that of square $\mathrm{ABCD}.$

In the given square $\mathrm{ABCD}, \frac{\mathrm{AQ}}{\mathrm{QD}}=\frac{\mathrm{DP}}{\mathrm{PC}}=\frac{4}{1}.$ Find the ratio of area of triangle $\mathrm{AOQ}$ to that of quadrilateral $\mathrm{OBCP}.$

Consider the following diagram in which $\square \mathrm{ABCD}$ is a rhombus and $\mathrm{AF}=\mathrm{EF}=\mathrm{EC}.$ A circle is inscribed in quadrilateral $\mathrm{AECF}$ and a circle is inscribed in quadrilateral $\mathrm{ABCD}.$ Each of the circles is of the largest possible area.

Find the ratio of the area of circle that lies in $\square \mathrm{AECF}$ to the area of the circle inscribed in $\square \mathrm{ABCD}.$

Area of the trapezium $\mathrm{ABCD}$ is $350 \mathrm{sq}. \mathrm{cm}$ and $\mathrm{AB}=21 \mathrm{cm},$ $\mathrm{CD}=49 \mathrm{cm}.$ The two diagonals bisect each other at $\mathrm{O},$ find the area of $∆\mathrm{AOD}.$

In the following diagram $\mathrm{ABCD}$ is a square. The mid-points $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}$ are connected in such a way that two congruent kites $\mathrm{SAPC}$ and $\mathrm{CRAQ}$ are formed. Area of each kite is $200 \mathrm{sq}. \mathrm{cm}.$ Find the area of common region contained by the two kites.