In the triangle ABC, it is known that the angle C is 90 degrees, AB is 9 cm, BC is 3 cm.

In the triangle ABC, it is known that the angle C is 90 degrees, AB is 9 cm, BC is 3 cm. On the hypotenuse AB, point M is marked so that AM: MB is 1: 2. Find CM.

Let the length of the segment AM = X cm, then, by condition, the length of the segment BM = 2 * X cm.

AM + BM = AB.

X + 2 * X = 9.

3 * X = 9.

X = 9/3 = 3.

AM = 3 cm, BM = 2 * 3 = 6 cm.

Let us define the cosine of the angle ABC in the right-angled triangle ABC.

CosABS = BC / AB = 3/9 = 1/3.

In the BCM triangle, we apply the cosine theorem for the triangle.

CM ^ 2 = BC ^ 2 + BM ^ 2 – 2 * BC * BM * CosB.

CM ^ 2 = 9 + 36 – 2 * 3 * 6 * 1/3 = 45 – 12 = 33.

CM = √33 cm.

Answer: The length of the CM segment is √33 cm.