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: Determine the coordinates of B. The coordinates of point B are given from the previous problem. The coordinates of B are (-2; -2). Step 2: Determine the equation of BQ. Q is the midpoint of AC. The coordinates of A are (-7; 8) and C are (5; -1). The midpoint formula is M = ((x_1 + x_2)/(2), (y_1 + y_2)/(2)). Q = ((-7 + 5)/(2), (8 + (-1))/(2)) Q = ((-2)/(2), (7)/(2)) Q = (-1, (7)/(2)) Now, find the equation of the line passing through B(-2; -2) and Q(-1; (7)/(2)). First, calculate the gradient m_BQ: m_BQ = (y_2 - y_1)/(x_2 - x_1) = (7)/(2) - (-2)-1 - (-2) = (7)/(2) + (4)/(2)-1 + 2 = (11)/(2)1 = (11)/(2) Using the point-slope form y - y_1 = m(x - x_1) with point B(-2; -2): y - (-2) = (11)/(2)(x - (-2)) y + 2 = (11)/(2)(x + 2) Multiply by 2 to clear the fraction: 2(y + 2) = 11(x + 2) 2y + 4 = 11x + 22 2y = 11x + 18 The equation of BQ is 11x - 2y + 18 = 0. Step 3: AM is a median of ABC. Determine the equation of AM. M is the midpoint of BC. The coordinates of B are (-2; -2) and C are (5; -1). M = ((-2 + 5)/(2), (-2 + (-1))/(2)) M = ((3)/(2), (-3)/(2)) Now, find the equation of the line passing through A(-7; 8) and M((3)/(2); -(3)/(2)). First, calculate the gradient m_AM: m_AM = (y_2 - y_1)/(x_2 - x_1) = (-3)/(2) - 8(3)/(2) - (-7) = (-3)/(2) - (16)/(2)(3)/(2) + (14)/(2) = (-19)/(2)(17)/(2) = -(19)/(17) Using the point-slope form y - y_1 = m(x - x_1) with point A(-7; 8): y - 8 = -(19)/(17)(x - (-7)) y - 8 = -(19)/(17)(x + 7) Multiply by 17: 17(y - 8) = -19(x + 7) 17y - 136 = -19x - 133 19x + 17y - 136 + 133 = 0 19x + 17y - 3 = 0 The equation of AM is 19x + 17y - 3 = 0. Step 4: Determine the coordinates of the centroid of ABC. The centroid G of a triangle with vertices (x_A, y_A), (x_B, y_B), and (x_C, y_C) is given by the formula: G = ((x_A + x_B + x_C)/(3), (y_A + y_B + y_C)/(3)) Using A(-7; 8), B(-2; -2), and C(5; -1): G_x = (-7 + (-2) + 5)/(3) = (-9 + 5)/(3) = (-4)/(3) G_y = (8 + (-2) + (-1))/(3) = (8 - 3)/(3) = (5)/(3) The coordinates of the centroid are (-(4)/(3); (5)/(3)). That's 2 down. 3 left today — send the next one.