In the triangle ABC, the segment BM is the median and BH is the height. It is known that AC = 40 and BC = BM. Find AH.

By condition, ВM is the median of the ABC triangle, since the median divides the side in half, then AM = CM = AC / 2 = 40/2 = 20 cm.

Consider the MBC triangle, in which the sides BC = BM, therefore the MBC triangle is isosceles. Then the height BH of the triangle MBС is also the median of the triangle, which means that it divides the segment MC in half. MH = CH = MS / 2 = 20/2 = 10 cm.

Let us determine the length of the segment AH. AH = AC – CH = 40 – 10 = 30 cm.

Answer: The length of the segment AH is 30 cm.