# Sunday Times Teaser – Field for Thought

### by Peter Good

#### Published Sunday August 16 2020 (link)

Farmer Giles had a rectangular field bordered by four fences that was 55 hectares in size. He divided the field into three by planting two hedges, from the mid-point of two fences to two corners of the field. He then planted two more hedges, from the mid-point of two fences to two corners of the field. All four hedges were straight, each starting at a different fence and finishing at a different corner.

What was the area of the largest field when the four hedges had been planted?

The layout used is shown above.  Since the answer doesn’t depend on shape of the rectangular field, we can derive the answer by considering it to be a square.  All triangles are similar with non-hypotenuse side ratios of 2:1. Using BC as a unit of length, triangles ABC and ADE are similar with ADE being double the size of ABC. So the side CE of the central square field is 2 and the whole field has a side length AD of $$2\sqrt{5}$$. Hence AD/BC = $$1/\sqrt{5}$$, which means that the area ratio of the inner and outer squares is $$1/5$$ making the central field 11 hectares. The red figures show the relative areas of all the sub-fields.