Polygons

$ABCD$ is a trapezium with bases $|AB|=8\mathrm{cm}$ and $|CD|=4\mathrm{cm}$. Calculate the area of the triangle $ABS$ if the area of the triangle $CDS$ is $12{\mathrm{cm}}^2$, where $S$ is the intersection point of the diagonals $BD$ and $AC$.

The profile of a trench has the shape of an isosceles trapezium, as shown in the picture. Determine the width of the trench.

Let $ABCD$ be a trapezium with the base $AB$ of $8\mathrm{cm}$. The remaining sides have the same length. The measure of $\measuredangle DAB$ is ${60}^{\circ }$. Calculate the perimeter of the trapezium.

The area of a rectangular trapezium is $35{\mathrm{cm}}^2$. The bases have lengths of $6\mathrm{cm}$ and $8\mathrm{cm}$. Express the measure of the angle between the longer base and the longer arm of the trapezium. Round the result to one decimal place.