In ΔABC, AB = 16 in, BC = 9 in, AC = 10 in. AD is perpendicular to the extension of BC. Find CD.

The Law of Cosines is useful for this. It tells you

b² = a² + c² - 2ac*cos (B)

where the length BD is c*cos (B). Solving for BD, we get

(b² - a² - c²) / (-2a) = BD

However, BD = BC + CD = a + CD, so we really want to find

CD = (a² + c² - b²) / (2a) - a

CD = (c² - a² - b²) / (2a)

Substituting the given numbers, we have

CD = (16² - 9² - 10²) / (2*9) = 75/18 = 25/6

The length CD is 25/6 = 4 1/6.