2006 AMC 10A 第 16 题

Let $O$ and $O'$ be the centers; the right triangles cut off along $AC$ are similar

The centers lie $3$ and $6$ from $A,$ and the base is $8$ below $A$

Let $O, O'$ be the centers of the small and large circles, and let $D$ be the point where the small circle touches $AC.$ The right triangles cut off along $AC$ are similar, so $\frac{AO}{1} = \frac{AO + 3}{2},$ giving $AO = 3$ and $AO' = 6.$

The tangent length is $AD = \sqrt{AO^2 - 1^2}$ $= \sqrt{3^2 - 1^2}$ $= 2\sqrt2.$ Let $F$ be the midpoint of $BC$; then $AF = AO' + 2 = 8.$

Since $\triangle ADO \sim \triangle AFC,$ we get $\frac{FC}{1} = \frac{AF}{2\sqrt2} = \frac{8}{2\sqrt2} = 2\sqrt2.$ Thus $BC = 4\sqrt2,$ and the area is $\frac12 \cdot BC \cdot AF$ $= \frac12 \cdot 4\sqrt2 \cdot 8$ $= 16\sqrt2.$

Thus, the correct answer is D.