Consider triangle \(\triangle ABC\) in the picture below, with \(\angle A = 30^{\circ}, \angle C= 45^{\circ}\) and \(AB=10\):
What are \(AC\) and \(BC\), in that order?
+2666 Draw altitude \(BM\).
\(BM\) splits \(\triangle ABC\) into 2 triangles: \(30-60-90\) and a \(45 - 45 - 90\) triangle.
10 is the hypotenuse of the \(30 - 60 - 90\) triangle, meaning \(BM = 5\), and \(AM = 5\sqrt3\)
\(\triangle BCM \) has the angles \(45 - 45- 90\) , so \(CM = 5\), and \(\color{brown}\boxed{BC = 5\sqrt2}\)
We also know \(AC = AM + MC\). Subbing in the known values, we find that \(\color{brown}\boxed{AC = 5 \sqrt3 + 5}\)
