In a right-angled triangle ABC Ð A = 90 °, AB = 20 cm, height AD is 12 cm.Find AC and cos C.

Consider the triangle ADB, by the Pythagorean theorem we find the leg DB.
DB² = AB² – AD² = 400 – 144 = 256, DB = 16 cm.
Triangles ADB and ABC are similar (two angles are equal). From the similarity of triangles it follows:
DB / AB = AB / BC
BC = AB * AB / DB = 20 * 20/16 = 25 cm.
By the Pythagorean theorem:
AC² = BC² – AB² = 625 – 400 = 225, AC = 15 cm.
Cos C = AC / BC = 15/25 = 3/5.
Answer: AC = 15 cm, cos C = 3/5.