Solution 4.2:9

From Förberedande kurs i matematik 1

If we introduce the dashed triangle below, the distance as the crow flies between A and B is equal to the triangle's hypotenuse, c.

One way to determine the hypotenuse is to know the triangle's opposite and adjacent sides, since the Pythagorean theorem then gives

In turn, we can determine the opposite and adjacent by introducing another triangle APR, where R is the point on the line PQ which the dashed triangle's side of length a cuts the line.

Because we know that AP=4 and the angle at P, simple trigonometry shows that x and y are given by

xy=4sin30=421=2=4cos30=423=23.

We can now start to look for the solution. Since x and y have been calculated, we can determine a and b by considering the horizontal and vertical distances in the figure.

a=x+5=2+5=7		b=12−y=12−23

With a and b given, the Pythagorean theorem leads to

c=a2+b2=72+(12−23)2=49+(122−21223+(23)2)=205−38311.0 km.