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
The area of the circle passing through the vertices P, Q, and R is approximately 32.0 cm². Here's how to calculate it: 1. Find the third angle of the triangle (∠R): The sum of angles in a triangle is 180°. ∠R = 180° - (∠P + ∠Q) ∠R = 180° - (50° + 60°) ∠R = 180° - 110° ∠R = 70° 2. Use the Law of Sines to find the circumradius (R): The Law of Sines states that for a triangle with sides a, b, c and opposite angles A, B, C, the ratio a/sin A = b/sin B = c/sin C is equal to 2R, where R is the circumradius. We have side PQ (let's call it 'r') = 6 cm, and its opposite angle is ∠R = 70°. r / sin(R) = 2R 6 / sin(70°) = 2R R = 6 / (2 * sin(70°)) R = 3 / sin(70°) 3. Calculate the value of R: sin(70°) ≈ 0.9396926 R ≈ 3 / 0.9396926 R ≈ 3.192537 cm 4. Calculate the area of the circle: The area of a circle is given by the formula A = πR². A = π * (3.192537)² A = π * 10.192297 A ≈ 3.14159265 * 10.192297 A ≈ 31.9599 cm² Rounding to three significant figures, the area is 32.0 cm².