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
Here is the solution to question 12: a) The sum of the opposite angles of a cyclic quadrilateral is 180^. b) Step 1: Identify the relationship between opposite angles in a cyclic quadrilateral. In a cyclic quadrilateral, the sum of opposite angles is 180^. Given WXY = x and WZY = 3x. Step 2: Set up the equation and solve for x. WXY + WZY = 180^ x + 3x = 180^ 4x = 180^ x = (180^)/(4) x = 45^ The value of x is 45^. c) This is an experimental task and cannot be performed by an AI. The theorem states that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore, the central angle WOY is twice the circumference angle WXY. WOY = 2 WXY d) Step 1: Use the value of x found in part (b) to determine WZY. From part (b), x = 45^. So, WZY = 3x = 3(45^) = 135^. Step 2: Identify properties of triangles formed by radii. Since O is the center of the circle, OW, OY, and OZ are radii. In OYZ, OY = OZ (radii of the same circle). Therefore, OYZ is an isosceles triangle, and its base angles are equal: OYZ = OZY In OWZ, OW = OZ (radii of the same circle). Therefore, OWZ is an isosceles triangle, and its base angles are equal: OWZ = OZW Step 3: Express WZY in terms of the angles from the isosceles triangles. The angle WZY of the cyclic quadrilateral is the sum of WZO and OZY. WZY = WZO + OZY Step 4: Substitute known values and relationships to prove the statement. We know WZY = 135^. So, WZO + OZY = 135^. Substitute WZO with OWZ (since WZO = OZW = OWZ) and OZY with OYZ (since OZY = OYZ). OWZ + OYZ = 135^ Hence, it is proven that OWZ + OYZ = 135^.