2005 AMC 12B Problem 18

Angle $A$ acute means $C$ is on the far side of the line through $A$ perpendicular to $AB$; similarly for $B$

Angle $C$ acute means $C$ lies outside the circle with diameter $AB$; combine three regions

Line $AB$ has slope $1.$ For $\angle A$ to be acute, $C$ must lie beyond the line through $A$ perpendicular to $AB;$ in the first quadrant that line runs between $P(4, 0)$ and $Q(0, 4).$ For $\angle B$ to be acute, $C$ must lie before the line through $B$ perpendicular to $AB,$ between $S(14, 0)$ and $T(0, 14).$

For $\angle C$ to be acute, $C$ must lie outside the circle $U$ with diameter $AB,$ whose radius is $\dfrac{AB}{2} = \dfrac{5\sqrt2}{2}.$

The circle lies entirely inside this strip and in the first quadrant. Thus the region is the large right triangle $OST$ minus the small right triangle $OPQ$ and the full circle $U:$ $$\begin{aligned} &\dfrac12 \cdot 14^2 - \dfrac12 \cdot 4^2 \\ &\quad {}- \pi\left(\dfrac{5\sqrt2}{2}\right)^2 \\ &= 98 - 8 - \dfrac{25\pi}{2} \\ &= 90 - \dfrac{25\pi}{2} \approx 51. \end{aligned}$$

Thus, the correct answer is C.