The segments AB and DC lie on parallel lines, and the segments AC and BD intersect at point M. Find MC if AB = 14, DC = 56, AC = 40.

1. Consider triangles DMC and AMB:
– angle DMC = angle AMB (since these angles are vertical and vertical angles are equal);
– angle CDM = angle ABM (since these angles are cross-lying, formed at the intersection of two parallel lines (AB and DC) secant (BD)).
Thus, two angles of triangle DMC are equal to two angles of triangle AMB, therefore, these triangles are similar.
2. Since DMC and AMB are similar triangles, then:
AM / MC = AB / DC;
AM + MC = AC.
Let’s substitute the known values ​​into these equalities:
AM / MS = 14/56;
AM + MC = 40.
A system of equations with two unknowns is obtained.
In the second equation, we express AM through MС:
AM = 40 – MС.
We substitute the resulting expression into the first equation of the system:
(40 – MС) / MС = 14/56;
56 (40 – MС) = 14MС (proportional);
2240 – 56MС = 14MС;
70MС = 2240;
MС = 2240/70;
MС = 32 cm.
Answer: MС = 32 cm.