Bearings Problem Solving Question

Question

I need help on this question where I don't know how to use bearings to answer this question. Please can have the step by step workings to answer this question.

 

 

Keiran runs 5km on a bearing of 030 degrees. He then runs 3km on a bearing of 270 degrees. He finally walks back to his starting point.

 

How far is his total journey?

What area has his journey enclosed?

 

 

Thank You and help is appreciated

Answer 1

1. The description of the solution would greatly benefit if a figure/diagram could be drawn, but since I know of no easy way to do that here, we'll need to rely on a purely written description
2. Picture a compass where north points vertically.  East is to the right, etc.  On a cartesian coordinate system, North would be the positive vertical axis.
3. Bearing refers to the angle that is made with this vertical y axis, in a clockwise direction.
4. Kieran intially runs 5km on a bearing of 30 degrees.  So he starts at the origin of the cartesian coordinate system, the pt (0,0), and travels at 30 degrees measured from the y axis.  This is equivalent to 60 degrees from the positive x axis.  A 30-60-90 triangle is formed.
5. When he finishes this part of his journey, he will be at a point on the coordinate system.  With a 30-60-90 triangle, the shortest side is half the hypotenuse.  The hypotenuse is 5, so the x coordinate is 2.5.  The longer leg is sqrt(3) * 2.5.  To be exact, you can should continue to leave this longer leg value as sqrt(3) * 2.5, but I'm going to use a calculator to estimate the value at 4.33
6. For the second part of his run, draw a new coordinate system where the pt (2.5, 4.33) is now the origin.  So "north" becomes the vertical line x = 2.5
7. 270 degrees means that he is traveling due West.  Since he is going 3km due West, he winds up at the same y coordinate, but the x coordinate is 2.5- 3 = -0.5.  So, at the end of the second run, he is at the point (-0.5, 4.33)
8. Although the problem doesn't specifically state it, we must assume that when he walks back to his origin, he is taking the shortest path.  He's not simply wandering about aimlessly until he gets back to home base.
9. To get back to the starting point, he simply travels the line that connects the point (-0.5, 4.33) and (0,0).
10. This walking distance W is given by W = sqrt (0.5² + 4.33²) =  4.36
11.  So the total journey is 5km + 3km + 4.36km =  12.36km
12. The triangle that is formed by his journey has a height of 4.33 and a base of 2.5+0.5 = 3. 
13. Area of a triangle is 0.5*b*h =  0.5 * 3 * 4.33 =  6.54 km²