I need to solve this trig problem. Can you please help me?
Based on this image:
I need to calculate $PO$ based on the values of $\alpha$, $\beta$ and $AB$ ( Assume that I know the values of $\alpha$ and $\beta$ angles and length of $AB$). $O$ is the middle of $AB$.
Can this problem be solved?
Any clue on solving this problem is very appreciating!
We make use of the Length of a Median formula relating the length of $PO$ with the lengths of the sides of the triangle. It is as follows $$(PA)^2+(PB)^2=2(PO)^2+\frac{(AB)^2}{2}$$
But the lengths $PA, PB$ are not given, so we need to calculate them.
To do so, we apply the law of sines, relating the lengths of the sides of the triangle $APB$ to the sines of its angles.
Let $\gamma$ be the internal angle of $APB$ on $P$. We have $$\frac{\sin \beta}{PA}=\frac{sin (\pi- \alpha)}{PB}=\frac{\sin \gamma}{AB}$$
Thus we have all the information needed to find $PO$.