I stop while out walking and take the bearing of a windmill and note it as θ º. I then walk 5 km north and take the bearing again – it is now 2 θ. How far away, in km, was the windmill from the position where I took the first bearing?

Answer:

19) C

A bearing is an angle measured clockwise from the north direction.
It is much easier to draw a diagram to show the situation.

The original bearing is ‘ø’ and when the person walks 5 km towards the north, the bearing becomes 2ø.

The angle inside the triangle adjacent to the 2ø, must be equal to 180º – 2ø

The sum angles inside the triangle which are opposite to the 2ø must be equal to 2ø. (exterior angle rule)

Since the initial bearing was ø, the third angle = 2ø – ø = ø

Since there are two angles which are the same in a triangle, the triangle must be isosceles.
2 equal sides = 5 km

Let the third side = T

Split the isosceles triangle in to two right-angled triangles. The third side must be 0.5 T

cos ø = adjacent = 0.5T
hypotenuse 5

5 cos ø = 0.5 T

T = 10 cos ø