Perpendicular to the height BD of the triangle ABC, the conduction intersects the side AB and BC at the point

Perpendicular to the height BD of the triangle ABC, the conduction intersects the side AB and BC at the point M and P. Find AB and the ratio of the rows of triangles MRB and ABC if it is known that MB = 7 BP = 9 PC = 18

Triangles ABC and MBP are similar, since the angle A is equal to the angle M, and the angle C is equal to the angle P. So, you can write down the ratio of their sides:

AB / MB = BC / BP.

Find aircraft:

BC = BP + PC = 9 + 18 = 27.

Find AB:

AB / 7 = 27/9;

AB = 27 * 7/9 = 21.

Area ratio of triangles:

S (ABC) / S (MBP) = k², where k is the coefficient of similarity of triangles. Find k:

k = AB / MB = 21/7 = 3;

S (ABC) / S (MBP) = 3² = 9.

Answer: side AB is 21, the ratio of the areas of triangles (ABC to MBP) is 9.