1. **Problem statement:** We have a rectangle ABCD with side lengths 10 cm by 10 cm. Two quarter circles of radius 5 cm are cut out from opposite corners, leaving a shaded figure with area 110 cm^2. We need to find the area of the original rectangle ABCD. 2. **Formula and rules:** - Area of rectangle: $$\text{Area}_{\text{rect}} = \text{length} \times \text{width}$$ - Area of a quarter circle: $$\text{Area}_{\text{quarter circle}} = \frac{1}{4} \pi r^2$$ - The shaded area is the area of the rectangle minus the areas of the two quarter circles. 3. **Calculate the area of the quarter circles:** - Radius $$r = 5$$ cm - Area of one quarter circle: $$\frac{1}{4} \pi (5)^2 = \frac{1}{4} \pi 25 = \frac{25\pi}{4}$$ cm^2 - Area of two quarter circles: $$2 \times \frac{25\pi}{4} = \frac{50\pi}{4} = \frac{25\pi}{2}$$ cm^2 4. **Set up the equation for the shaded area:** $$\text{Area}_{\text{shaded}} = \text{Area}_{\text{rect}} - \text{Area}_{\text{two quarter circles}}$$ Given $$\text{Area}_{\text{shaded}} = 110$$ cm^2, $$110 = \text{Area}_{\text{rect}} - \frac{25\pi}{2}$$ 5. **Solve for the area of the rectangle:** $$\text{Area}_{\text{rect}} = 110 + \frac{25\pi}{2}$$ 6. **Calculate numerical value:** Using $$\pi \approx 3.1416$$, $$\frac{25\pi}{2} = \frac{25 \times 3.1416}{2} = \frac{78.54}{2} = 39.27$$ So, $$\text{Area}_{\text{rect}} = 110 + 39.27 = 149.27$$ cm^2 **Final answer:** The area of rectangle ABCD is approximately $$149.27$$ cm^2.