Side AB of triangle ABC is 16cm. Angle A = 30 °. angle B = 105 °. Calculate the length of the BC side.

Let’s draw the height BО. The ABO triangle is rectangular. If two angles are known, <A = 30, <O = 90, then <ABO = 180 – <A – <O. <ABO = 180 – 30 – 90 = 60.

In a right-angled triangle, the leg opposite the 30-degree angle is half the hypotenuse.

BО = AB / 2; BO = 16/2 = 8 (cm).

<CBO = <ABC – <ABO; <CBO = 105 – 60 = 45

In the BOC triangle, <C = 180 – <BOC – <CBO; <C = 45. So the triangle BOС is isosceles, BO = OС = 8 cm.

Let us apply the Pythagorean theorem for the BOS triangle.

BC ^ 2 = BO ^ 2 + OC ^ 2;

BC ^ 2 = 8 ^ 2 + 8 ^ 2 = 2 * 64; BC = 8√2 cm.

Answer. BC = 8√2 cm.