The length of the segment AB is 50cm. Points M and N lie on this segment. Find the length

The length of the segment AB is 50cm. Points M and N lie on this segment. Find the length of the segment MN if: 1) AM = 23cm, NB = 21cm; 2) AN = 42cm, MB = 34cm.

Decision.
It is given that AB = AM + MN + NB.
1. By the condition of the problem AB = 50 cm, AM = 23 cm, NB = 21 cm.
Then 50 = 23 + MN + 21, whence MN = 50 – 23 – 21 = 50 – 44 = 6 (cm).
Answer. MN = 6 cm.
2. According to the problem statement AN = 42 cm, MB = 34 cm.
Initial data: AB = AM + MN + NB can be written in a different way, through the segments AN and MB, the equality will not change, it will remain true: AB = AN + MB – MN.
Then 50 = 42 + 34 – MN, whence MN = 42 + 34 – 50 = 76 – 50 = 26 cm.
Answer. MN = 26 cm.