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Of All the Areas

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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?