In triangular ABC, point K is marked on its median BM so that BK: KM = 7: 3.

In triangular ABC, point K is marked on its median BM so that BK: KM = 7: 3. Find the ratio of the area of triangle ABK to the area of triangle ABC.

We introduce the proportionality coefficient x and we get that ВK = 7x, KM = 3x.
The median BM divides the ABC triangle into two equal triangles, from this it follows:
S ABM = 1/2 S ABC
S ABM = S ABK + S AKM.
In triangle ABM we will draw the height АН.
S ABK = 1/2 * ВK * AН = 1/2 * 7x * AН;
S AKM = 1/2 * KM * AН = 1/2 * 3x * AН;
S ABM = 1/2 * 7x * AН + 1/2 * 3x * AН = 1/2 * 10x * AН = 5x * AН.
S ABC = 2 * S ABM = 10x * AH.
Find the ratio of the areas of triangles ABK and ABC:
S ABK / S ABC = 1/2 * 7x * AH / 10x * AH = 7/20.
Answer: The areas of triangles ABK and ABC are related as 7/20.