1. **State the problem:** Calculate the area of the shaded region formed by a large quarter circle of radius 12, a smaller quarter circle of radius 5, and a right triangle with legs 12 and 5 inside a rectangle. 2. **Formula and rules:** - Area of a quarter circle: $$\frac{\pi r^2}{4}$$ - Area of a right triangle: $$\frac{1}{2} \times \text{leg}_1 \times \text{leg}_2$$ - The shaded area = Area of big quarter circle - Area of triangle - Area of small quarter circle 3. **Calculate each area:** - Big quarter circle radius $r=12$: $$\frac{3.14 \times 12^2}{4} = \frac{3.14 \times 144}{4} = \frac{451.68}{4} = 112.92$$ - Triangle area with legs 12 and 5: $$\frac{1}{2} \times 12 \times 5 = 30$$ - Small quarter circle radius $r=5$: $$\frac{3.14 \times 5^2}{4} = \frac{3.14 \times 25}{4} = \frac{78.5}{4} = 19.625$$ 4. **Calculate shaded area:** $$112.92 - 30 - 19.625 = 63.295$$ 5. **Final answer:** The area of the shaded region is **63.295** square units.