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.
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Here is the solution for question 5 from the image. 5. Find the length of the following pair of point The distance formula between two points (x_1, y_1) and (x_2, y_2) is given by: D = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) (1) (-3,4) and (-5,0) Step 1: Identify the coordinates. Let (x_1, y_1) = (-3, 4) and (x_2, y_2) = (-5, 0). Step 2: Substitute the coordinates into the distance formula. D = sqrt((-5 - (-3))^2 + (0 - 4)^2) D = sqrt((-5 + 3)^2 + (-4)^2) D = sqrt((-2)^2 + 16) D = sqrt(4 + 16) D = sqrt(20) Step 3: Simplify the radical. D = sqrt(4 × 5) D = 2sqrt(5) The final answer for 5(1) is 2sqrt(5). (2) (2,0) and (3,4) Step 1: Identify the coordinates. Let (x_1, y_1) = (2, 0) and (x_2, y_2) = (3, 4). Step 2: Substitute the coordinates into the distance formula. D = sqrt((3 - 2)^2 + (4 - 0)^2) D = sqrt((1)^2 + (4)^2) D = sqrt(1 + 16) D = sqrt(17) The final answer for 5(2) is sqrt(17). (3) ((1)/(2), 1) and (1, (1)/(2)) Step 1: Identify the coordinates. Let (x_1, y_1) = ((1)/(2), 1) and (x_2, y_2) = (1, (1)/(2)). Step 2: Substitute the coordinates into the distance formula. D = sqrt((1 - (1)/(2))^2 + ((1)/(2) - 1)^2) D = sqrt(((2)/(2) - (1)/(2))^2 + ((1)/(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: Rationalize the denominator. D = (1)/(sqrt(2)) × sqrt(2)sqrt(2) D = sqrt(2)2 The final answer for 5(3) is sqrt(2)2.