Calculate the semi-perimeter of triangle PQR.

Step 1: Calculate the semi-perimeter of triangle PQR. The side lengths are PQ = c = 7.8 cm, PR = b = 6.6 cm, and QR = a = 5.9 cm. The semi-perimeter s is given by: s = (a+b+c)/(2) s = (5.9 + 6.6 + 7.8)/(2) s = (20.3)/(2) s = 10.15 cm Step 2: Calculate the area of triangle PQR using Heron's formula. The area K is given by: K = sqrt(s(s-a)(s-b)(s-c)) K = sqrt(10.15(10.15-5.9)(10.15-6.6)(10.15-7.8)) K = sqrt(10.15(4.25)(3.55)(2.35)) K = sqrt(359.0068125) K ≈ 18.94748 cm^2 a) Step 3: Calculate the radius of the circumcircle. The radius R of the circumcircle is given by the formula: R = (abc)/(4K) R = (5.9 × 6.6 × 7.8)/(4 × 18.94748) R = (304.788)/(75.78992) R ≈ 4.0215 cm Correct to one decimal place: R ≈ 4.0 cm The radius of the circle is 4.0 cm. b) Step 4: Calculate the angles of the triangle using the Cosine Rule. For angle P (opposite side a=5.9 cm): P = (b^2 + c^2 - a^2)/(2bc) P = (6.6^2 + 7.8^2 - 5.9^2)/(2 × 6.6 × 7.8) P = (43.56 + 60.84 - 34.81)/(102.96) P = (69.59)/(102.96) ≈ 0.67599 P = (0.67599) ≈ 47.47^ For angle Q (opposite side b=6.6 cm): Q = (a^2 + c^2 - b^2)/(2ac) Q = (5.9^2 + 7.8^2 - 6.6^2)/(2 × 5.9 × 7.8) Q = (34.81 + 60.84 - 43.56)/(92.04) Q = (52.09)/(92.04) ≈ 0.56595 Q = (0.56595) ≈ 55.53^ For angle R (opposite side c=7.8 cm): R = (a^2 + b^2 - c^2)/(2ab) R = (5.9^2 + 6.6^2 - 7.8^2)/(2 × 5.9 × 6.6) R = (34.81 + 43.56 - 60.84)/(77.88) R = (17.53)/(77.88) ≈ 0.22509 R = (0.22509) ≈ 77.00^ The angles of the triangle are P ≈ 47.5^, Q ≈ 55.5^, R ≈ 77.0^. c) Step 5: Calculate the area of the shaded region. The area of the circle is A_circle = R^2. Using the more precise value of R ≈ 4.0215 cm: A_circle = (4.0215)^2 A_circle = (16.17246225) A_circle ≈ 50.806 cm^2 The area of