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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 Pythagoras' theorem then gives

c2=a2+b2

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

\displaystyle \begin{align} & x=4\sin 30^{\circ }=4\centerdot \frac{1}{2}=2, \\ & y=4\cos 30^{\circ }=4\centerdot \frac{\sqrt{3}}{2}=2\sqrt{3} \\ \end{align}

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

\displaystyle a=x+5=2+5=7

\displaystyle b=12-y=12-2\sqrt{3}

With a and \displaystyle b given, Pythagoras' theorem leads to

\displaystyle \begin{align} & c=\sqrt{a^{2}+b^{2}}=\sqrt{7^{2}+\left( 12-2\sqrt{3} \right)^{2}} \\ & =\sqrt{49+\left( 12^{2}-2\centerdot 12\centerdot 2\sqrt{3}+\left( 2\sqrt{3} \right)^{2} \right)} \\ & =\sqrt{205-38\sqrt{3}}\quad \approx \quad 11.0\quad \text{km}\text{.} \\ \end{align}