Since AD is the height of the ABC triangle, then the ABD triangle is rectangular, then, according to the Pythagorean theorem, AD ^ 2 = AB ^ 2 – BD ^ 2 = 225 – 144 = 81.
ВD = 9 cm.
Height AD is drawn from a right angle to the hypotenuse, then AD ^ 2 = BD * CD.
СD = АD ^ 2 / ВD = 144/9 = 16 cm.
Then the side length ВС = ВD + СD = 9 + 16 = 25 cm.
By the Pythagorean theorem, AC ^ 2 = BC ^ 2 – AB ^ 2 = 625 – 225 = 400.
AC = 20 cm.
CosACB = AC / BC = 20/25 = 4/5.
Answer: The length of the AC side is 20 cm, the cosine of the C angle is 4/5.
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