In the triangle ABC BM is the median and BH is the height. It is known that AC = 53 and BC = BM find AH.

On the segment AC, the points are located in the sequence: A, M, H, C.
Angle A is less than angle C.
(1) AH = AC – HC = 53 – HC.
Let’s find the segment HС.
Since BM is the median, AM = MC = AC / 2.
And since BC = BM by condition, the MBC triangle is isosceles, and ВН is its height and median.
Therefore, HC = MH = MC / 2 = AM / 2 = AC / 4 = 53/4 = 13.25.
Substitute the value of HC in equation (1).
AH = 53 – HC = 53 – 13.25 = 39.75.

Answer: the segment AH is 39.75.