In a right-angled triangle ABC, the angle between the bisector CK and the height CH

In a right-angled triangle ABC, the angle between the bisector CK and the height CH, drawn from the vertex of the right angle C, is 15 degrees. Side AB = 14cm. Find side AC if it is known that point K lies between B and H.

The CK bisector divides the right angle C in half, which means that the angle ВСК = 90/2 = 45 degrees. The KCH angle between the CK bisector and the CH height is 15 degrees. The angle BCH is equal to the sum of the angles ВСК and КСH: the angle ВСH = 45 + 15 = 60 degrees. Because CH – height, then triangle ВHС – rectangular with hypotenuse ВС. The sum of the angles of a triangle is 180, which means angle B = 180-angle BHC-angle BH = 180-90-60 = 30 degrees. The ratio of the opposite leg to the hypotenuse is equal to the sine of the angle, which means sinB = AC / AB, hence AC = AB * sinB = AB * sin30 = 14 * 0.5 = 7 cm.