A right-angled triangle ABC is given. The bisector CP is dropped from the vertex of the right angle C to the hypotenuse.

A right-angled triangle ABC is given. The bisector CP is dropped from the vertex of the right angle C to the hypotenuse. Find the length of the leg AC if AB = 15 and the angle CPB = 75 degrees.

Consider a triangle CPВ: the bisector CP divides the right angle C in half, so the angle PCB is 45 degrees. According to the condition of the problem, the angle CPВ is 75 degrees, the sum of the angles of the triangle is 180, we can find the angle B: 180-PCV-CPV = 180-45-75 = 60 degrees. The cosine of the angle is the ratio of the opposite leg to the hypotenuse, cosB = AC / AB. Hence AC = AB * cosB = 15 * cos60 = 15 * 0.5 = 7.5.