+980 In triangle $ABC$, $\angle A = 30^\circ$ and $\angle B = 60^\circ$. Point $X$ is on side $\overline{AC}$ such that line segment $\overline{BX}$ bisects $\angle ABC$. If $AB = 12$, then find the area of triangle $BXA$.
In triangle $ABC,$ point $D$ is on $\overline{AC}$ such that $AD = 3CD = 12$. If $\angle ABC = \angle BDA = 90^\circ$, then what is $BD$?
Find the area of triangle $PQR$ if $PQ = QR = 12$ and $\angle PQR = 120^\circ$.
In triangle $ABC,$ $AB = 10,$ $BC = 24,$ and $AC = 26.$ Find the length of the shortest altitude in this triangle.
First problem: By the angle bisector theorem, the area of triangle BXA is 15*sqrt(2).
Second problem: BD^2 = AD*CD = 3*12 = 36, so BD = 6.
