Can someone help me find "x" from the diagram

My teacher told me these kinds of problems were going to be on the test. Can you please tell me step by step how you get this answer.

http://i1175.photobucket.com/albums/r621…

From figure, we have,
AC = 10, AD = x, DB = 21,
From Δ ACD : x/10 = cos(A) ........................................… [1]
Also,
From Δ ABC : 10 / (x+21) = cos(A) ............................. [2[,

FROm [1] & [2], x/10 = 10/(x+21),
OR,
x(x+21) = 100, ===> x^2 + 21x - 100 = 0,
OR,
(x-4)(x+25) = 0,
OR,
x = -25 .... negative ... neglected,
AND,
x = 4 >=======================================… ANSWER

Just tell me this: Are you using Pythagorean theorem, or are you using Sin, Cos, Tan, and so on?