In triangle ABC, angle A is 90 degrees, angle B is equal to 60 degrees. on the AC side, point D is marked

In triangle ABC, angle A is 90 degrees, angle B is equal to 60 degrees. on the AC side, point D is marked so that the angle DBC is 30 degrees, DA is 4 cm. Find AC and the distance from point D to BC.

Angle ABC = 60, and angle DBC = 30, then segment BD is the bisector of angle B, and angle ABD is also equal to 30.

In a right-angled triangle ABD, the hypotenuse BD = 2 * AD = 2 * 4, since AD lies opposite the angle 30.

DН is perpendicular to BC, since the angle ВDН = 30, and ВD = 8 cm, then DH = ВD / 2 = 8/2 = 4 cm In a right-angled triangle СDН the angle DСН = (90 – 60) = 30, then СD = 2 * DH = 2 * 4 = 8 cm.

Side length АС = АD + СD = 4 + 8 = 12 cm.

Answer: The AC side is 12 cm, from point D then the straight BC is 4 cm.