This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Determine the angle of the sector. The figure shows a triangle OAB where OA and OB are radii of the circle, so OA = OB = 5.00 cm. The length of the chord AB is given as 8.66 cm. We can use the Cosine Rule to find the angle = AOB. The Cosine Rule states c^2 = a^2 + b^2 - 2ab C. In triangle OAB: AB^2 = OA^2 + OB^2 - 2(OA)(OB) () (8.66)^2 = (5)^2 + (5)^2 - 2(5)(5) () 74.9956 = 25 + 25 - 50 () 74.9956 = 50 - 50 () 74.9956 - 50 = -50 () 24.9956 = -50 () () = (24.9956)/(-50) = -0.499912 This value is very close to -0.5. Therefore, = (-0.5) = 120^. To use in the sector area formula, convert to radians: = 120^ × ()/(180^) = (2)/(3) radians Step 2: Calculate the area of the sector OAB. The area of a sector is given by the formula A_sector = (1)/(2) r^2 (where is in radians) or A_sector = ()/(360^) r^2 (where is in degrees). Using degrees: A_sector = (120^)/(360^) × (5)^2 A_sector = (1)/(3) × 25 A_sector = (25)/(3) cm^2 A_sector ≈ (25 × 3.14159265)/(3) ≈ 26.1799 cm^2 Step 3: Calculate the area of the triangle OAB. The area of a triangle can be calculated using the formula A_triangle = (1)/(2) ab C. A_triangle = (1)/(2) (OA)(OB) () A_triangle = (1)/(2) (5)(5) (120^) A_triangle = (25)/(2) × sqrt(3)2 A_triangle = 25sqrt(3)4 cm^2 A_triangle ≈ (25 × 1.73205081)/(4) ≈ 10.8253 cm^2 Step 4: Calculate the area of the shaded region. The shaded region is a segment of the circle, which is the area of the sector minus the area of the triangle. A_segment = A_sector - A_triangle A_segment = (25)/(3) - 25sqrt(3)4 A_segment = 25 (()/(3) - sqrt(3)4) A_segment ≈ 26.1799 - 10.8253 A_segment ≈ 15.3546 cm^2 Rounding to two decimal places: A_segment ≈ 15.35 cm^2 The final answer is 15.35 cm^2. 3 done, 2 left today. You're making progress.