In a right-angled triangle ABC, angle A = 90 AB = 20 AD = 12 (height) find AC and cosC.

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

BH ^ 2 = AB ^ 2 – AH ^ 2 = 400 – 144 = 256.

BH = 16 cm.

AH is the height drawn from a right angle to the hypotenuse, then the square of its length is equal to the product of the segments into which the height divides the hypotenuse.

AH ^ 2 = BH * CH.

CH = AH ^ 2 / BH = 144/16 = 9 cm.

From the right-angled triangle ACН, according to the Pythagorean theorem, we determine the length of the hypotenuse AC.

AC ^ 2 = AH ^ 2 + CH ^ 2 = 144 + 81 = 225.

AC = 5 cm.

Then CosC = CH / AC = 9/15 = 3/5.

Answer: The length of the AC segment is 15 cm, CosC = 3/5.