In a triangle ABC, the angle C is 90 degrees, CH is the height, AH = 12, cos A = 2/3. Find AB.

The same angle A is common for two triangles – a large one and one of the small ones.

Find the AC side of the triangle. At the same time, AC is the hypotenuse in the triangle AHC. Find through the cosine of the angle A (CAH):

cos A = AH / AC;

2/3 = 12 / AC;

AC = 18;

Now we find the hypotenuse AB of triangle ABC, where AC will act as a leg. Find through the cosine of the same angle A (BAC):

cos A = AC / AB;

2/3 = 18 / AB;

AB = 27.

Answer: 27.