In triangle ABC the midpoints M and N of sides BC and AC are marked, respectively.

In triangle ABC the midpoints M and N of sides BC and AC are marked, respectively. The area of triangle CNM is 2. Find the area of quadrilateral ABMN.

Triangles CNM and ABC are similar because they have a common angle C, and the sides forming an angle C are 1: 2.

MC / BC = 1/2,

NC / AC = 1/2.

The areas of similar triangles ABC and MCN are related as the squares of the ratios of the corresponding sides:

SCNM / SABC = NC2 / AC2;

2 / SABC = 1/4;

SABC = 8 cm2.

The area of the quadrilateral ABMN is equal to the difference between the areas of triangles ABC and CNM:

SABMN = SABC – SCNM = 8 cm2 – 2 cm2 = 6 cm2.

Answer: 6 cm2.