The segments AB and DC lie on parallel lines, and the segments AC and BD intersect

The segments AB and DC lie on parallel lines, and the segments AC and BD intersect at point M. Find the MC if AB = 15, DC = 30, AC = 39.

Consider the resulting triangles ABM and CDM. These triangles are similar, since all three angles in them are respectively equal:

<AMB = <DMC, as vertical angles, <MAB = <MCD, <MBA = <MDA, as angles lying internally with parallel lines AB and CD. Let’s move on to the dimensions of the sides, using the property of the sides in similar triangles:

DC / AB = MC / AM = 30/15 = 2, AM + MC = AC = 39.

MC = 2 * AM, AC = 2 * AM + AM = 3 * AM = 39. AM = 39/3 = 13.

MC = 2 * AM = 2 * 13 = 26.

Check: DC / AB = MC / AM = 30/15 = 2 = 26/13 = 2.

Answer: MS = 26.