In the ABC triangle, the AC side = 12 HM is the median, BH is the height, BC = BM. Find the length of the segment AH.

1. By the condition of the problem ВС = ВМ, that is, the ВСМ triangle is isosceles.

2. АМ = СМ = 1/2 АС, since the median ВМ divides АС into equal segments АМ and СМ.

AM = CM = 12: 2 = 6 centimeters.

3. According to the properties of an isosceles triangle, the height of the BH drawn to its base

MC, also performs the functions of the median, that is, it divides the base into two the same segment of CH and MH. Therefore CH = MH = 6: 2 = 3 centimeters.

4. Calculate the length of the segment AH:

AH = AM + MH = 6 + 3 = 9 centimeters.

Answer: the length of the segment AH is 9 centimeters.