In triangle ABC, angle C is 90o, cos A = 4/5, AC = 4. Find the height CH.

Let us write by definition what is the cosine of angle A in a given triangle:
Cos A = AC / AB. We find the hypotenuse AB:
AB = AC / cos A = 4 / 4/5 = 5.
Let us find, according to the Pythagorean theorem, the second unknown leg BC of the triangle ABC:
ВС = √ (AB² – AC²) = √ (5² – 4²) = √ (25 – 16) = √9 = 3.
Consider two triangles ABC and AСН. They are similar (right angle and angle A are common). From the similarity of triangles, we write down the aspect ratio:
CH / AC = BC / AB
CH = AC * BC / AB = 4 * 3/5 = 12/5 = 2.4.
Answer: the CH height is 2.4.