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.

A few of the sides of a regular pentagon are extended in such a way that it forms an isosceles triangle as shown in the given diagram. If the perimeter of the pentagon is $10 \mathrm{cm},$ what is the perimeter of the isosceles triangle $\mathrm{APQ} ?$

A few of the sides of a regular pentagon are extended in such a way that it forms an isosceles triangle as shown in the given diagram. If the perimeter of the pentagon is $10 \mathrm{cm},$ what is the perimeter of the isosceles triangle $\mathrm{PQR} ?$

In the following figure, if the area of the shaded region (the smaller pentagon) is $\left(\frac{7-3\sqrt{5}}{2}\right) \mathrm{sq}. \mathrm{cm},$ find the area of the larger regular pentagon.

In the following figure when all the five diagonals of a regular pentagon are drawn they form another small pentagon at the centre which is shown as a shaded region. If each side of the larger (original) pentagon measures $1$ inch, and $\phi =\frac{\sqrt{5}+1}{2},$ find the ratio of the area of larger pentagon to the area of smaller pentagram in terms of $\phi .$