Points MN are marked on the segment AB so that AM = 9 cm, BN = 8 cm.Find the length of the segment MN if AB = 11 cm.

In order to determine the location of points M and N on the segment AB, pay attention to the fact that AM = 9 cm. Then since AM + MB = AB, then MB = AB – AM = 11 cm – 9 cm = 2 cm. This means that point M is located close to point B than to point A. Now we find the location of point N. Arguing similarly, we have AN + BN = AB. Then AN = AB – BN = 11 cm – 8 cm = 3 cm. Hence, point N is located closer to point A than to point B. So, the order of the points is as follows: from left to right – A, N, M, B.
Using the location of the points, we can assert that AN + MN + MB = AB, from which we can easily find the desired length of the segment MN = AB – AN – MB = 11 cm – 3 cm – 2 cm = 6 cm.
Answer: MN = 6 cm.