# Sunday Times Teaser 2485

### by Nick MacKinnon

#### Published: 9 May 2010 (link)

Jack and Kay [1] have inherited a circular field, with North Gate (N) at the northernmost point and East Gate (E), South Gate (S) and West Gate (W) at the appropriate points.

Jack’s share of the field is one hectare in area. For each point P in his share, three of the angles NPE, EPS, SPW and WPN are acute.

For each point in Katy’s [1] share, however, fewer than three of those angles are acute.

How far is it between North Gate and East Gate?

[1] There is a Sunday Times typographic error here – the intended name is is not known.

One Comment Leave one →
We can calculate this area as the area of the field ($$\pi r^2$$) minus eight times the area of the orange coloured segment. And the orange area is one quarter of a circle of radius $$r/\sqrt{2}$$ minus the area of a triangle of the width and height equal to $$r/\sqrt{2}$$. So the green area is:
$\pi r^2 – 8(\pi r^2/8 – r^2/4) = 2r^2$
Hence $$2r^2 = 10000$$ square metres, which gives $$r$$ as $$100/\sqrt{2}$$ metres. Hence the distance between the north and east gates ($$\sqrt{2}r$$) is 100 metres.