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
To find the length between two points (x_1, y_1) and (x_2, y_2), we use the distance formula: D = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) i) For the points (-3, 4) and (-5, 0): Let (x_1, y_1) = (-3, 4) and (x_2, y_2) = (-5, 0). Step 1: Substitute the coordinates into the distance formula. D = sqrt((-5 - (-3))^2 + (0 - 4)^2) Step 2: Simplify the terms inside the square root. D = sqrt((-5 + 3)^2 + (-4)^2) D = sqrt((-2)^2 + (-4)^2) D = sqrt(4 + 16) D = sqrt(20) Step 3: Simplify the square root. D = sqrt(4 × 5) = 2sqrt(5) The length is 2sqrt(5). ii) For the points (2, 0) and (3, 4): Let (x_1, y_1) = (2, 0) and (x_2, y_2) = (3, 4). Step 1: Substitute the coordinates into the distance formula. D = sqrt((3 - 2)^2 + (4 - 0)^2) Step 2: Simplify the terms inside the square root. D = sqrt((1)^2 + (4)^2) D = sqrt(1 + 16) D = sqrt(17) The length is sqrt(17). iii) For the points ((1)/(2), 1) and (1, 1(1)/(2)): First, convert 1(1)/(2) to an improper fraction: 1(1)/(2) = (3)/(2). So the points are ((1)/(2), 1) and (1, (3)/(2)). Let (x_1, y_1) = ((1)/(2), 1) and (x_2, y_2) = (1, (3)/(2)). Step 1: Substitute the coordinates into the distance formula. D = sqrt((1 - (1)/(2))^2 + ((3)/(2) - 1)^2) Step 2: Simplify the terms inside the square root. D = sqrt(((2)/(2) - (1)/(2))^2 + ((3)/(2) - (2)/(2))^2) D = sqrt(((1)/(2))^2 + ((1)/(2))^2) D = sqrt((1)/(4) + (1)/(4)) D = sqrt((2)/(4)) D = sqrt((1)/(2)) Step 3: Simplify the square root. D = sqrt(1)sqrt(2) = (1)/(sqrt(2)) To rationalize the denominator: D = (1)/(sqrt(2)) × sqrt(2)sqrt(2) = sqrt(2)2 The length is sqrt(2)2.