In a right-angled triangle ABC, angle C is 90 degrees, CB = 20cm, height CD = 12cm. Find AC and AB.

In a right-angled triangle ВСD, according to the Pythagorean theorem, we determine the length of the leg ВD.

ВD ^ 2 = ВС ^ 2 – СD ^ 2 = 400 – 144 = 256.

ВD = 16 cm.

Since the height CD is drawn from the top of the right angle to the hypotenuse, then CD ^ 2 = BD * AD.

AD = CD ^ 2 / BD = 144/16 = 9 cm.

Then AB = AD + BD = 9 + 16 = 25 cm.

In a right-angled triangle ACD, according to the Pythagorean theorem, we determine the length of the hypotenuse AC.

AC ^ 2 = AD ^ 2 + CD ^ 2 = 81 + 144 = 225.

AC = 15 cm.

Answer: The length of the AC is 15 cm, the length of AB is 25 cm.