Fresh day Pablo, let's solve. Step 1: Identify the geometry and given values. The two circles have the same radius, $r = 8.75$ cm. The common chord PQ has a length of $8.75$ cm. The area common to the two circles consists of two identical circular segments. Step 2: Determine the angle subtended by the chord at the center of one circle. Consider one circle with center $O_1$. The radii $O_1P$ and $O_1Q$ are both equal to the radius $r = 8.75$ cm. The common chord PQ is also $8.75$ cm. Therefore, the triangle $\triangle O_1PQ$ has sides $O_1P = O_1Q = PQ = 8.75$ cm. This means $\triangle O_1PQ$ is an equilateral triangle. The angle subtended by the chord PQ at the center $O_1$ is $\theta = \angle PO_1Q = 60^\circ$. In radians, $\theta = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3}$ radians. Step 3: Calculate the area of the sector $O_1PQ$. The area of a sector is given by the formula $A_{sector} = \frac{1}{2}r^2\theta$. $$A_{sector} = \frac{1}{2}(8.75)^2\left(\frac{\pi}{3}\right)$$ $$A_{sector} = \frac{76.5625\pi}{6} \text{ cm}^2$$ Step 4: Calculate the area of the triangle $O_1PQ$. Since $\triangle O_1PQ$ is an equilateral triangle with side length $s = 8.75$ cm, its area is given by $A_{triangle} = \frac{\sqrt{3}}{4}s^2$. $$A_{triangle} = \frac{\sqrt{3}}{4}(8.75)^2$$ $$A_{triangle} = \frac{76.5625\sqrt{3}}{4} \text{ cm}^2$$ Step 5: Calculate the area of one circular segment. The area of one circular segment is the area of the sector minus the area of the triangle. $$A_{segment} = A_{sector} - A_{triangle}$$ $$A_{segment} = \frac{76.5625\pi}{6} - \frac{76.5625\sqrt{3}}{4}$$ $$A_{segment} = 76.5625 \left( \frac{\pi}{6} - \frac{\sqrt{3}}{4} \right) \text{ cm}^2$$ Step 6: Calculate the total common area. The common area to the two circles is composed of two such identical segments. $$A_{common} = 2 \times A_{segment}$$ $$A_{common} = 2 \times 76.5625 \left( \frac{\pi}{6} - \frac{\sqrt{3}}{4} \right)$$ $$A_{common} = 76.5625 \left( \frac{2\pi}{6} - \frac{2\sqrt{3}}{4} \right)$$ $$A_{common} = 76.5625 \left( \frac{\pi}{3} - \frac{\sqrt{3}}{2} \right) \text{ cm}^2$$ Now, substitute the numerical values: $$A_{common} = 76.5625 \left( \frac{3.14159265}{3} - \frac{1.73205081}{2} \right)$$ $$A_{common} = 76.5625 (1.04719755 - 0.86602540)$$ $$A_{common} = 76.5625 (0.18117215)$$ $$A_{common} \approx 13.86809 \text{ cm}^2$$ Rounding to two decimal places: $$A_{common} \approx 13.87 \text{ cm}^2$$ The area common to the two circles is $\boxed{\text{13.87 cm}^2}$. Send me the next one 📸