Example - (2004 American Invitational Mathematics Examination) A circle of radius 1 is randomly placed inside a $15 \times 36$ rectangle $ABCD$. Find the probability that this circle does not intersect with the diagonal $AC$.
Analysis We use the area relationship of the figures.
Solution If a circle with radius 1 is placed inside $ABCD$, its center must lie within a $13 \times$ 34 rectangle, with an area of 442. If the circle does not intersect with the diagonal $AC$, then the center of the circle lies within two right-angled triangular regions (the shaded parts in Figure 14-1), where the hypotenuse is 1 unit away from $AC$. The area of the shaded parts can be calculated as 375. Therefore, $P=\frac{375}{442}$.