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 are the solutions to Question 3: 3.1 Determine the co-ordinate of M the midpoint of line RT. To find the midpoint M of a line segment with endpoints (x_1, y_1) and (x_2, y_2), we use the formula: M = ((x_1 + x_2)/(2), (y_1 + y_2)/(2)) Given R(-5, 3) and T(6, 6). Step 1: Substitute the coordinates of R and T into the midpoint formula. M = ((-5 + 6)/(2), (3 + 6)/(2)) Step 2: Calculate the coordinates of M. M = ((1)/(2), (9)/(2)) The coordinates of M are ((1)/(2), (9)/(2)). 3.2 Calculate the gradient of the line WT. To calculate the gradient m of a line passing through two points (x_1, y_1) and (x_2, y_2), we use the formula: m = (y_2 - y_1)/(x_2 - x_1) Given W(-3, 0) and T(6, 6). Step 1: Substitute the coordinates of W and T into the gradient formula. m_WT = (6 - 0)/(6 - (-3)) Step 2: Simplify the expression to find the gradient. m_WT = (6)/(6 + 3) m_WT = (6)/(9) m_WT = (2)/(3) The gradient of line WT is (2)/(3). 3.3 Prove that RW is perpendicular to WT. For two lines to be perpendicular, the product of their gradients must be -1. We already have m_WT = (2)/(3). Now we need to calculate the gradient of RW. Given R(-5, 3) and W(-3, 0). Step 1: Calculate the gradient of RW. m_RW = (0 - 3)/(-3 - (-5)) m_RW = (-3)/(-3 + 5) m_RW = (-3)/(2) Step 2: Multiply the gradients of RW and WT. m_RW × m_WT = (-(3)/(2)) × ((2)/(3)) m_RW × m_WT = -1 Since the product of their gradients is -1, RW is indeed perpendicular to WT. 3.4 If RWTK is a parallelogram, determine the values of a and b. In a parallelogram, the diagonals bisect each other, meaning they share a common midpoint. The diagonals of RWTK are RT and WK. We already found the midpoint of RT in 3.1 as M_RT = ((1)/(2), (9)/(2)). Now we find the midpoint of WK using W(-3, 0) and K(a, b). Step 1: Set up the midpoint formula for WK. M_WK = ((-3 + a)/(2), (0 + b)/(2)) Step 2: Equate the x-coordinates of M_RT and M_WK. (-3 + a)/(2) = (1)/(2) -3 + a = 1 a = 1 + 3 a = 4 Step 3: Equate the y-coordinates of M_RT and M_WK. (0 + b)/(2) = (9)/(2) b = 9 The values are a = 4 and b = 9. Last free one today — make it count tomorrow, or type /upgrade for unlimited.