1. **Problem statement:** We have a square ABCD with side length $14.5$ cm. Point B is the center of a quarter circle arc passing through points D and C. The highlighted area is the region inside the square bounded by the arc and the sides AB and BC. We want to find the area of this highlighted region. 2. **Understanding the figure:** - ABCD is a square with side length $s = 14.5$ cm. - The quarter circle is centered at B, with radius equal to the side length of the square, $r = 14.5$ cm. - The quarter circle arc passes through points D and C. - The highlighted area is the quarter circle sector minus the right triangle formed by points B, C, and the foot of the perpendicular from C to AB. 3. **Formula for the quarter circle area:** The area of a full circle is $\pi r^2$. A quarter circle area is: $$\text{Area}_{\text{quarter circle}} = \frac{1}{4} \pi r^2$$ 4. **Calculate the quarter circle area:** $$\text{Area}_{\text{quarter circle}} = \frac{1}{4} \pi (14.5)^2 = \frac{1}{4} \pi \times 210.25 = 52.5625 \pi$$ 5. **Calculate the area of the right triangle inside the square:** The triangle is formed by points B (center), C (top-right corner), and the foot of the perpendicular from C to AB (which lies on AB). The height from C to AB is given as 9 cm. - The base of the triangle is $14.5$ cm (side AB). - The height is $9$ cm. Area of triangle: $$\text{Area}_{\triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 14.5 \times 9 = 65.25$$ 6. **Calculate the highlighted area:** The highlighted area is the quarter circle sector minus the triangle: $$\text{Area}_{\text{highlighted}} = 52.5625 \pi - 65.25$$ 7. **Numerical approximation:** Using $\pi \approx 3.1416$: $$52.5625 \times 3.1416 \approx 165.1$$ So, $$\text{Area}_{\text{highlighted}} \approx 165.1 - 65.25 = 99.85 \text{ cm}^2$$ **Final answer:** The area of the highlighted region is approximately $99.85$ square centimeters.