Question

Sarah leaves her house and walks 11 miles to the west and then 6 miles at an angle 30 degrees west of north. How far and in what direction does Sarah need to walk to get home?

A picture and detailed show of work would be appreciated!

Answer 1

Sarah first walks from C to B (11 miles) and then from B to A (6 miles). Her path home should be along the line AC, exactly joining her starting point C to the ending point A. To find the length AC, note that the triangle ADC is a right angled triangle with hypotenuse AC. Also note that triangle ADB is right angled with AB the hypotenuse. Thus, we can find AD and DB, and then using AD and DC the length AC by utilising the Pythagoras theorem.

AD = ABcos30 = 6cos30 = 5.19 mi

DB = ABsin30 = 6sin30 = 3 mi

Thus, DC = DB + BC = 3 + 11 = 14 mi

Thus, using the Pythagoras theorem: AD² + DC² = AC²

i.e. AC = √(AD²+AC²) = √(5.19²+14²) =√(26.93+196)

= √222.93 = 14.93 mi

Now we need to find the angle DAC to find the direction she must go in.

Let this angle be represented by a. Therefore, tan(a) = DC/AD = 14/5.19

Thus, a = arctan (14/5.19) = 69.65°.

Note, that the direction she must walk in is thus 69.65° east of South, and the distance she must walk is 14.93 miles.