In the triangle ABC, the angles ABC = 60 degrees and ACB = 90 degrees are known, and point D

In the triangle ABC, the angles ABC = 60 degrees and ACB = 90 degrees are known, and point D splits the hypotenuse into parts AD = 1 and DB = 3. What is the length of the segment CD.

Hypotenuse AB = AD + DB = 1 + 3 = 4.
The BC leg is adjacent to the ABC corner. The ratio of the adjacent leg to the hypotenuse is the cosine of the angle:
cosABC = BC / AB;
BC = AB * cosABC = 4 * cos60 = 4 * 0.5 = 2.
In the triangle BCD, we know the sides BD and BC and the angle between them DBC, which coincides with the angle ABC, equal to 60 degrees. Therefore, the square of the side CD can be found by the cosine theorem as the sum of the squares of the sides BD and BC minus the double product of these sides by the cosine of the angle between them:
CD ^ 2 = BD ^ 2 + BC ^ 2-2 * BD * BC * cosB = 3 ^ 2 + 2 ^ 2-2 * 2 * 3 * cos60 = 9 + 4-2 * 2 * 3 * 0.5 = 13-6 = 7;
CD = √7.