In a triangle ABC AC = BC = 12, tgA = √2 / 4. Find the height of the CH.

Since AC = BC by condition, triangle ABC is isosceles with apex at C. Then, by the definition of the tangent, we get:

CH = AH * tg (A) = AH * √2 / 4.

By the Pythagorean theorem, we obtain the equality:

CH ^ 2 + AH ^ 2 = CA ^ 2;

AH ^ 2 * 16/2 + AH ^ 2 = 144;

AH ^ 2 = 144/9;

AH = 12/3 = 4.

Answer: the value of the height AH is 4.