A triangle ABC with a perimeter of 42 cm, one side is divided by a point M and divides the triangle by a segment

A triangle ABC with a perimeter of 42 cm, one side is divided by a point M and divides the triangle by a segment BM into two triangles with a perimeter of 32 and 35 cm, respectively, find the length of the segment BM.

1. P (perimeter) Δ ABC = AB + BC + AC = 42 centimeters.

P Δ ABM = AB + AM + BM = 32 centimeters.

P Δ ВСМ = ВС + СМ + ВМ = 35 centimeters.

2. Add the second and third expressions:

AB + AM + BM + BC + CM + BM = 67 centimeters.

3. AC = AM + CM.

4. Substitute instead of AC = AM + CM and AB + BC + AC = 42 in the resulting expression:

42 + 2 BM = 67 centimeters.

2BM = 67 – 42 = 25 centimeters.

BM = 25: 2 = 12.5 centimeters.

Answer: BM = 12.5 centimeters.