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Dividing the Field

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two trapeziums each of equal area. How could he do this?

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Same Height

A trapezium is divided into four triangles by its diagonals. Suppose the two triangles containing the parallel sides have areas a and b, what is the area of the trapezium?

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Major Trapezium

A small circle in a square in a big circle in a trapezium. Using the measurements and clue given, find the area of the trapezium.

Towering Trapeziums

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

$OGH$ is an isosceles right-angled triangle:
 
Lines $AB$, $CD$, $EF$, and $GH$ are parallel.

Suppose the area of the smallest triangle $OAB$ is one square unit.
  • If lines $OC$ and $AB$ have the same length, calculate the area of trapezium $ABDC$.
  • If lines $OE$ and $CD$ also have the same length, calculate the area of trapezium $CDFE$.
  • If lines $OG$ and $EF$ also have the same length, calculate the area of trapezium $EFHG$.
Suppose that the chain of trapezia continued. 

What would be the area of the $n^{th}$ trapezium in the chain?

Can you explain your results?