I have an isosceles triangle $ABC$, where the height $h$ and angle at vertex $A$ are known. The Cartesian coordinates of vertex $A$ are also known to be $\left(x,y\right)$.
If the angle between the y-axis and the line represented by $h$ is $θ$, is it possible to find the Cartesian coordinates of points $B$ and $C$?
Best Answer
Absolutely.
The angle between the $y$-axis and $AB$ is $\theta - \tfrac{A}{2}$, and the angle between the $y$-axis and $AC$ is $\theta + \tfrac{A}{2}$. So you will get \begin{align*} B &= (x - r \sin(\theta - \tfrac{A}{2}), y + r \cos(\theta - \tfrac{A}{2}) \\ C &= (x - r \sin(\theta + \tfrac{A}{2}), y + r \cos(\theta + \tfrac{A}{2}) \end{align*} where $r = AB = AC = \frac{h}{\cos(\tfrac{A}{2})}$.