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.
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