Geometry Question (Tips)

Question

My question is do I use the formula $a^2+b^2 = c^2$ or do I have to solve it with something else like for example trigonometry.

Answer 1

By law of cosines we obtain: $$x=\sqrt{140^2+180^2-2\cdot140\cdot180\cdot\cos135^{\circ}}.$$ I got $x\approx296.037$

Answer 2

As Michael Rozenberg stated, the Law of Cosines gives you your answer.

The formula you mentioned in your question is the special case when the angle in between the two sides is a right angle:

$$c^2 = a^2 + b^2 - 2ab \cos 90^{\circ} = a^2 + b^2 - 2ab(0) = a^2 + b^2.$$

Answer 3

Only by Pythagorean theorem, we can observe that

• total west displacement $w=180+140\cdot \frac{\sqrt 2}2$
• total nord displacement $n=140\cdot \frac{\sqrt 2}2$

then

$$x=\sqrt{w^2+n^2}=\sqrt{180^2+140^2\frac12+\sqrt 2 \cdot180\cdot 140+140^2 \frac12}=\sqrt{180^2+140^2+\sqrt 2 \cdot180\cdot 140}\approx296$$