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

In the triangle ABC, the midpoints M and H of the sides BC and AC, respectively, are marked. The area of the triangle CHM is 12. Find the area of the quadrangle ABMH.

Decision:

Triangles ABC and HMC are similar. They have three angles equal, the ABC angle is equal to the HMC angle, the BAC angle is equal to the MHC angle. Since side MH is the midline of triangle ABC and it is parallel to side AB. The coefficient of similarity of these triangles is 2 since MC = BC / 2, HC = AB / 2 and MH = AB / 2.

Theorem: the ratio of the areas of two similar triangles is equal to the square of the similarity coefficient.

And so, if the area of the triangle HMC is 12 then the area of the triangle ABC is
2 ^ 2 * 12 = 48. The area of the quadrilateral ABMH is 48 – 12 = 36.

Answer: 36.