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: Calculate the gradient of AB. The coordinates of A are (-10, -2) and B are (10, 8). m_AB = (y_B - y_A)/(x_B - x_A) = (8 - (-2))/(10 - (-10)) = (10)/(20) = (1)/(2) Step 2: Determine the gradient of the altitude from C to AB. The altitude is perpendicular to AB. If two lines are perpendicular, the product of their gradients is -1. m_CD = -(1)/(m_AB) = -(1)/(1/2) = -2 Step 3: Find the equation of the altitude using point C (11, -9) and m_CD = -2. Using the point-gradient form y - y_1 = m(x - x_1): y - (-9) = -2(x - 11) y + 9 = -2x + 22 y = -2x + 22 - 9 y = -2x + 13 This matches the given equation. 3.1.2 Step 1: Find the equation of line AB. Using the gradient m_AB = (1)/(2) and point A (-10, -2): y - y_A = m_AB(x - x_A) y - (-2) = (1)/(2)(x - (-10)) y + 2 = (1)/(2)(x + 10) y + 2 = (1)/(2)x + 5 y = (1)/(2)x + 3 Step 2: Solve the system of equations for line AB and the altitude CD to find point D. Equation of AB: y = (1)/(2)x + 3 Equation of CD: y = -2x + 13 Set the y-values equal: (1)/(2)x + 3 = -2x + 13 Multiply by 2 to eliminate the fraction: x + 6 = -4x + 26 x + 4x = 26 - 6 5x = 20 x = 4 Substitute x = 4 into the equation for the altitude: y = -2(4) + 13 y = -8 + 13 y = 5 The coordinates of point D are (4, 5). 3.2 To calculate the area of ABC, we can use the formula: Area = (1)/(2) × base × height. The base is AB and the height is CD (the altitude from C to AB). Step 1: Calculate the length of AB. Using the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) for A (-10, -2) and B (10, 8): AB = sqrt((10 - (-10))^2 + (8 - (-2))^2)