Point A lies on line BC between points B and C. Find the length of the segment BC if AB: AC = 7: 2 and AB – AC = 10 cm.

Since AB: AC = 7: 2 and AB – AC = 10, then from these data it is possible to compose a system of equations:
AB / AC = 7/2;
AB – AC = 10.
In the second equation, we express AB through AC and substitute the resulting expression into the first equation:
AB = 10 + AC;
(10 + AC) / AC = 7/2.
Let’s solve the resulting equation with one unknown:
7AC = 2 (10 + AC);
7AC = 20 + 2AC;
7АС – 2АС = 20;
5AC = 20;
AC = 20/5;
AC = 4 cm.
Substitute the obtained value of AC into the second equation of the system of equations and find the value of AB:
AB – AC = 10;
AB – 4 = 10;
AB = 10 + 4;
AB = 14 cm.
The segment BC is equal to:
BC = AC + AB;
BC = 4 + 14 = 18 (cm).
Answer: BC = 18 cm.