Question

Alfred the Great leaves London and travels in the direction NW for 80km. He then
turns and travels in the direction South 65 degrees west
until he is due west of London.

How far is he from London now?

Answer 1

The first leg (A) of Alfred's journey takes him 40√2 km west of London and 40√2 km north of London because 80km is the length of the hypotenuse of a right isosceles triangle. The base of this triangle is heading due west from London. The second leg of the journey (B) takes him further west by 40√2tan65. This time we have a right triangle with a vertex angle of 65 degrees and the leg of that triangle is Alfred's distance north of London=40√2km.

So the total distance west of London is 40√2(1+tan65)=177.88km approx.