In triangle ABC, angle C = 90 degrees, CH-height, AH = 15, tg angle A = 1/5. Find the segment BH.

Since CH is the height, the triangles AСН and ВСН are rectangular.

In a right-angled triangle ACН, we determine the length of the kata CH.

CH = AH * tgBAC = 15 * 1/5 = 3 cm.

Let us prove the similarity of the triangles AСН and BCH.

Let the angle СВН = α0, then the angle BAC = (90 – α) 0. In a right-angled triangle ACН, the angle ACН = (90 – (90 – α)) = α0. Then the angle АСН = СВН, and the right-angled triangles АСН and ВСН are similar in acute angle.

From the similarity of the triangles it follows: AH / CH = CH / BН.

BH = CH ^ 2 / AH = 9/15 = 3/5 cm.

Answer: The length of the ВН segment is 3/5 cm.