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 the quadrilateral ABMN is 24. Find the area of the triangle CNM.

Given:
triangle ABC,
M middle sun,
N – the middle of the speaker,
S avmn = 24,
Find S сnм -?
Decision:
1) The ABC triangle is similar to the CNM triangle according to the second similarity feature, since the sides NC / AC = MC / BC = 1/2 and the angle C is common;
2) The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.
Then S cnm / S abc = 1/4;
3) S abc = S abmn + S cnm;
S cnm / S avmn + S cnm = 1/4;
S cnm / 24 + S cnm = 1/4;
4 * S cnm = 24 + S cnm;
3 * S cnm = 24;
S cnm = 24: 3;
S сnм = 8.
Answer: 8