In triangle ABC, angle C = 90, CH- height AB = 24, tgA = 1/7. find BH

Let’s use the definition of tangent and write down the ratio:
Tg A = BC / AC = 1/7.
Let’s define that the leg BC is equal to x, then AC is equal to 7x.
Let’s write down the Pythagorean theorem for a given triangle and find the square of the leg BC.
AB² = BC² + AC²
24² = x² + 49x²
50x² = 576
x² = 11.52
Knowing that the leg in a right-angled triangle is the average proportional between the hypotenuse and its projection onto the hypotenuse, we write:
ВС² = AB * BH, hence
BH = BC² / AB = 11.52 / 24 = 0.48.
Answer: the BH segment is equal to 0.48.