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: Let the centres be O and Q with OQ = 5 cm, radius OA = 3.5 cm, and radius QB = 2.5 cm, where A and B are the points of intersection. Let M be the midpoint of common chord AB, so AM = MB = m cm. The line OQ is perpendicular to AB at M. Let OM = p cm, so QM = 5 - p cm. Step 2: Apply Pythagoras theorem in OMA: OA^2 = OM^2 + AM^2 (3.5)^2 = p^2 + m^2 ((7)/(2))^2 = p^2 + m^2 (49)/(4) = p^2 + m^2 (1) Step 3: Apply Pythagoras theorem in QMB: QB^2 = QM^2 + MB^2 (2.5)^2 = (5 - p)^2 + m^2 ((5)/(2))^2 = (5 - p)^2 + m^2 (25)/(4) = (5 - p)^2 + m^2 (2) Step 4: Subtract equation (2) from equation (1): (49)/(4) - (25)/(4) = p^2 + m^2 - [ (5 - p)^2 + m^2 ] (24)/(4) = p^2 - (5 - p)^2 6 = p^2 - (25 - 10p + p^2) 6 = p^2 - 25 + 10p - p^2 6 = 10p - 25 10p = 31 p = (31)/(10) cm Step 5: Substitute p = (31)/(10) into equation (1): m^2 = (49)/(4) - ((31)/(10))^2 m^2 = (49)/(4) - (961)/(100) m^2 = (1225)/(100) - (961)/(100) = (264)/(100) = (66)/(25) cm^2 m = sqrt((66)/(25)) = sqrt(66)5 cm Step 6: Length of common chord AB = 2m: AB = 2 × sqrt(66)5 = 2sqrt(66)5 cm 2sqrt(66)5 cm