In triangle ABC, angle A = 30° and angle B = 60°. Point X is on side AC such that line segment BX bisects angle ABC. If the length of side AC = 12, then find the area of triangle BXA.
Since angle A = 30° and angle B = 60°, then angle C = 180° - 30° - 60° = 90°. Therefore, triangle ABC is a right triangle with right angle at C.
Since BX bisects angle ABC, then angle XCB = 30°/2 = 15°. Therefore, triangle BXC is a 30-60-90 triangle.
The length of BC is AC/2 = 12/2 = 6.
The length of BX is BCsqrt(3)/2 = 6sqrt(3)/2 = 3*sqrt(3).
The area of triangle BXA is (1/2)BXXC = (1/2)3sqrt(3)12 = 18sqrt(3).
So the answer is 18√3
