Find the Angle BAC

Question

AB,AC,BC and h are known and its a isosceles triangle how to find angle BAC?

Answer 1

The triangle is completely determined by its side lengths $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, which you say are all known (the knowledge of $h$ and $ABC$ being isosceles is unnecessary).

To find $\angle BAC$, apply the law of cosines to the side lengths:

$$\overline{BC}^2=\overline{AB}^2+\overline{AC}^2-2\times\overline{AB}\times\overline{AC}\times\cos\angle BAC$$

Answer 2

Let D be the point where the altitude touches segment BC. Then $\frac{AD}{AB}$=$\cos\theta$, or $\theta=\text {Angle BAD}=\arccos(\frac{AD}{AB})$. The same logic apples to angle CAD=$\arccos(\frac{AD}{AC}).$ Adding these 2 values gives the value of Angle CAB (Angle Addition Postulate.)