In a right-angled triangle ABC A = 90, AB = 20cm; Find AC and cosC if AD height is 12 cm.

The height AD has formed two right-angled triangles for us. Consider one of them, triangle ABD, find the unknown leg BD using the Pythagorean theorem.
BD = √ (AB² – AD²) = √ (400 – 144) = √256 = 16 cm.
Let us use the property of the height dropped to the hypotenuse from the right angle and find CD.
AD² = BD * CD
CD = AD² / BD = 144/16 = 9 cm.
In the triangle ABC we find the hypotenuse BC.
BC = BD + CD = 16 + 9 = 25 cm.
Find the AC leg:
AC = √ (BC² – AB²) = √ (625 – 400) = √225 = 15 cm.
Cos C = AC / BC = 15/25 = 0.6.
Answer: AC = 15 cm, cos C = 0.6.