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 are the solutions for the given problems. Part 1: Mid-points of line segments The formula for the midpoint M(x_m, y_m) of a line segment joining (x_1, y_1) and (x_2, y_2) is: x_m = (x_1 + x_2)/(2), y_m = (y_1 + y_2)/(2) a) For points (2,3) and (4,7) Step 1: Substitute the coordinates into the midpoint formula. x_m = (2 + 4)/(2) = (6)/(2) = 3 y_m = (3 + 7)/(2) = (10)/(2) = 5 The midpoint is (3,5). b) For points (-1,3) and (5,4) Step 1: Substitute the coordinates into the midpoint formula. x_m = (-1 + 5)/(2) = (4)/(2) = 2 y_m = (3 + 4)/(2) = (7)/(2) The midpoint is (2, (7)/(2)). c) For points (29,5) and (9,25) (Assuming '2b' is a typo for '25') Step 1: Substitute the coordinates into the midpoint formula. x_m = (29 + 9)/(2) = (38)/(2) = 19 y_m = (5 + 25)/(2) = (30)/(2) = 15 The midpoint is (19,15). Part 2: Finding coordinates of B using the section formula Given point A (6,8), point P (9,2), and the ratio AP:PB = 3:7. Let B be (x_2, y_2). The section formula for a point P dividing a segment AB in the ratio m:n is: x = (nx_1 + mx_2)/(m+n), y = (ny_1 + my_2)/(m+n) Here, (x_1, y_1) = (6,8), (x,y) = (9,2), m=3, n=7. Step 1: Set up the equation for the x-coordinate. 9 = (7(6) + 3(x_2))/(3+7) 9 = (42 + 3x_2)/(10) 90 = 42 + 3x_2 3x_2 = 90 - 42 3x_2 = 48 x_2 = (48)/(3) = 16 Step 2: Set up the equation for the y-coordinate. 2 = (7(8) + 3(y_2))/(3+7) 2 = (56 + 3y_2)/(10) 20 = 56 + 3y_2 3y_2 = 20 - 56 3y_2 = -36 y_2 = (-36)/(3) = -12 Step 3: State the coordinates of B. The coordinates of B are (16, -12). What's next?