# Given triangle ABC, the perimeter of which is 42 cm. Point M is taken on the AC side, so that the perimeter

**Given triangle ABC, the perimeter of which is 42 cm. Point M is taken on the AC side, so that the perimeter of triangle ABM and triangle BCM are 32 cm and 35 cm, respectively. Find the length of the segment BM.**

P (triangle ABC) = AB + BC + CA = 42 cm; also by condition it is specified that AC = AM + MS, because on the AC side the point M is taken; P (triangle ABM) = AB + BM + MA = 32 cm; P (triangle BMC) = BC + CM + MB = 35 cm; then P (triangle ABC) = P (triangle ABM) – MB + P (triangle BMC) – MB; Substitute the given values into the equations for the perimeter of the triangle ABC, the unknown side MB will be denoted by the variable x:

42 = 32 – x + 35 – x;

2x = 32 + 35 – 42;

2x = 67 – 42;

2x = 25;

x = 25: 2;

x = 12.5 (cm) – side BM.

Answer: BM = 12.5 cm.