In the figure, the midpoints of the diagonals of a rhombus are joined to form a small quadrilateral:

  

Prove that this quadrilateral is a rhombus.

In the figure, the midpoints of the diagonals of a rhombus are joined to form a small quadrilateral:

The area of the small rhombus is $3 \mathrm{square} \mathrm{centimetres}$. The area of the large rhombus is $\mathrm{k} {\mathrm{cm}}^2$. Find the value of $k$.

The area of the largest rhombus that can be drawn inside a rectangle of sides $6$ centimetres and $4$ centimetres is $\mathrm{K} {\mathrm{cm}}^2$. Find the value of $\mathrm{K}$.

Draw a rectangle of sides $7 \mathrm{centimetres}$ and $4 \mathrm{centimetres}$. Draw isosceles trapezium of the same area, with the following specifications.

Lengths of parallel sides $9 \mathrm{centimetres}$, $5 \mathrm{centimetres}$.

Draw a rectangle of sides $7 \mathrm{centimetres}$ and $4 \mathrm{centimetres}$. Draw isosceles trapezium of the same area, with the following specifications.

Lengths of non-parallel sides $5 \mathrm{centimetres}$.

The area of the isosceles trapezium drawn below is $\mathrm{K} {\mathrm{cm}}^2$. Find the value for $\mathrm{K}$.

The parallel sides of an isosceles trapezium are $8 \mathrm{centimetres}$ and $14 \mathrm{centimetres}$ long; and non-parallel sides are $5 \mathrm{centimetre}$ long. Its area is $\mathrm{K} {\mathrm{cm}}^2$. Find the value for $\mathrm{K}$.

The lengths of the parallel sides of a trapezium are $30 \mathrm{centimetres}$, $10 \mathrm{centimetres}$ and the distance between them is $20 \mathrm{centimetres}$. The area is $\mathrm{K} {\mathrm{cm}}^2$. Find the value for $\mathrm{K}$.

Compute the area of the trapezium shown below:

(Express the result in $\mathrm{sq}.\mathrm{cm}$).