In triangle ABC, angle A = 30 degrees angle B = 105 degrees AB = 12. Find the BC side.

1. Let’s draw the height BH. Consider a triangle ABН:

∠АНB = 90, ∠А = 30, therefore, ∠ABН = 180 – 90 – 30 = 60.

2. AB = 12 – hypotenuse; АН and ВН – legs. Since opposite an angle of 30 degrees lies a leg equal to half of the hypotenuse, BH = 6.

3. Consider the BCH triangle:

Since ∠АВН = 60, and ∠В = 105, then ∠НВС = 45; ∠BНС = 90, therefore, ∠BCН = 45. It follows from this that the triangle BНС is isosceles, right-angled.

4. Find the side BC by the Pythagorean theorem:

BC ^ 2 = 2 * 6 ^ 2

BC = 6√2.