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's the solution to the problem. a) Find the coordinates of the mid-points K of AB and P of AC. Step 1: Find the coordinates of K, the midpoint of AB. The midpoint formula for two points (x_1, y_1) and (x_2, y_2) is ((x_1+x_2)/(2), (y_1+y_2)/(2)). Given A(3, 4) and B(-7, 2): K = ((3 + (-7))/(2), (4 + 2)/(2)) K = ((-4)/(2), (6)/(2)) K = (-2, 3) Step 2: Find the coordinates of P, the midpoint of AC. Given A(3, 4) and C(1, -2): P = ((3 + 1)/(2), (4 + (-2))/(2)) P = ((4)/(2), (2)/(2)) P = (2, 1) The coordinates are K(-2, 3) and P(2, 1). b) Find the equation of the perpendicular bisector of the line KP. Step 1: Find the midpoint of the line segment KP. Let M be the midpoint of KP. Given K(-2, 3) and P(2, 1): M = ((-2 + 2)/(2), (3 + 1)/(2)) M = ((0)/(2), (4)/(2)) M = (0, 2) Step 2: Find the gradient of the line segment KP. The gradient formula for two points (x_1, y_1) and (x_2, y_2) is m = (y_2 - y_1)/(x_2 - x_1). m_KP = (1 - 3)/(2 - (-2)) m_KP = (-2)/(2 + 2) m_KP = (-2)/(4) m_KP = -(1)/(2) Step 3: Find the gradient of the perpendicular bisector. The product of the gradients of two perpendicular lines is -1. Let m_ be the gradient of the perpendicular bisector. m_KP · m_ = -1 -(1)/(2) · m_ = -1 m_ = -1 · (-2) m_ = 2 Step 4: Find the equation of the perpendicular bisector. Using the point-slope form y - y_1 = m(x - x_1) with midpoint M(0, 2) and gradient m_ = 2: y - 2 = 2(x - 0) y - 2 = 2x y = 2x + 2 The equation of the perpendicular bisector is y = 2x + 2. That's 2 down. 3 left today — send the next one.