Answer to Question #161262 in Trigonometry for libby

Question

On a walking holiday you travel from the start, 𝑆, 5km on a bearing of 60° to point 𝑃. Then you change direction and walk for 5km on a bearing of 300° arriving at the finish 𝐹. What is your distance and bearing now from the start? Draw a diagram as part of your answer. 

Answer 1

Let SX be the axe from which we count the bearing. It means angle PSX = 60 degrees. Let PX1 be the line parallel to SX. As the bearing is 300 degrees, the angle X1PF1 equals 360 - 300 = 60 degrees, where F1 is the intersection of line SX and vector PF (our second move after we turned around at point P). Lets notice that angle PF1S is also 60 degrees, because PX1 is parallel to SF1. Then triangle PSF1 has angles 60, 60 and 180 - (60 + 60) = 60 degrees, so it is an equilateral triangle. We know that SP = 5, so PF1 = 5. But we know that PF = 5, and vector PF lies on vector PF1, so F and F1 is the same point. That means that SF1 is the distance we are looking for, and it also equals 5, because PSF is an equilateral triangle. As one can see now, the bearing is 0 degrees, because F lies on SX.

Answer: distance = 5km, bearing = 0 degrees