Segment CH – the height of the triangle ABC with a right angle C. Find BC, AH and CH if AB = 20cm and AC = 16cm.

To solve this problem, recall the Pythagorean theorem: the square of the hypotenuse is equal to the sum of the squares of the legs. Knowing that the hypotenuse AB = 20 cm, and the leg AC = 16 cm, we calculate the leg CB.
CB ^ 2 = 20 ^ 2-16 ^ 2 = 20 * 20-16 * 16 = 400-256 = 144.
CB = √144 = 12 cm.
Now, using the height formula through the lengths of the sides, we calculate the height CH.
CH = AC * CB / AB = 16 * 12/20 = 192/20 = 9.6 cm.
Now, from the triangle AHC, by the Pythagorean theorem, we calculate the leg AH.
AH ^ 2 = AC ^ 2-CH ^ 2 = 16 * 16-9.6 * 9.6 = 256-92.16 = 163.84.
AH ^ 2 = √163.84 = 12.8 cm.
Answer: 12 cm, 12.8 cm, 16 cm.