Power of a Point 2

Question

Let $B$, $C$, and $D$ be points on a circle. Let $\overline{BC}$ and the tangent to the circle at $D$ intersect at $A$. If $AB = 4$, $AD = 8$, and $\overline{AC} \perp \overline{AD}$, then find $CD$.

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This looks like a Power of a Point problem but I don't know how to apply it. All solutions are greatly appreciated.

Answer 1

Well, you have $$AC.AB = AD^2$$, then $AC = \frac{8^2}{4} = 16$.

Now, you can easily compute $CD$: $CD^2 = AD^2 + AC^2 = 8^2 + 16^2 = 320$, then $CD = 8\sqrt{5}$.