In an isosceles obtuse triangle ABC with base AB, the height is AH = 4, BC = 4√5. Find tg ABC.

Since, by condition, the triangle ABC is isosceles, then AC = AB = 4 * √5 cm.

AH is the height of the triangle, then the triangle ACH is rectangular, in which, according to the Pythagorean theorem, we determine the length of the leg CH.

CH ^ 2 = AC ^ 2 – AH ^ 2 = (4 * √5) ^ 2 – 4 ^ 2 = 80 – 16 = 64.

CH = 8 cm.

Then BH = CH + BC = 8 + 4 * √5 = 4 * (2 + √5) cm.

The AВН triangle is rectangular.

Then tgАВС = tgАВН = АН / ВН = 4/4 * (2 + √5) = 1 / (2 + √5).

Answer: tgABС = 1 / (2 + √5).