Prove that for any quadrilateral with diagonals perpendicular, the area is half the product of the diagonals.

The area of the quadrilateral that shown below is $\mathrm{K} {\mathrm{cm}}^2$. Find the value of $\mathrm{K}$.

The area of the parallelogram shown below is $\mathrm{K} {\mathrm{cm}}^2$. Find the value of $\mathrm{K}$.

The three blue lines in the picture below are parallel:

Prove that the areas of the quadrilaterals $ABCD$ and $PQRS$ are in the ratio of the lengths of the diagonals $AC$ and $PR.$

How should the diagonals be related for the quadrilaterals to have equal area?

The three blue lines in the picture below are parallel:

Prove that the areas of the quadrilaterals $ABCD$ and $PQRS$ are in the ratio of the lengths of the diagonals $AC$ and $PR.$ 

Draw two quadrilaterals, neither parallelograms nor trapeziums, of area $15$ square centimetres.