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 the coordinate geometry problems: 2.1 Calculate the perimeter of ABCD. Leave your answer in surd form. A rhombus has four equal sides. We can calculate the length of one side, for example, AD, and then multiply it by 4 to find the perimeter. Given points A (-8, 6) and D (3, 9). Step 1: Calculate the length of side AD using the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). AD = sqrt((3 - (-8))^2 + (9 - 6)^2) AD = sqrt((3 + 8)^2 + (3)^2) AD = sqrt((11)^2 + 3^2) AD = sqrt(121 + 9) AD = sqrt(130) Step 2: Calculate the perimeter of the rhombus. Perimeter = 4 × AD Perimeter = 4sqrt(130) The perimeter of ABCD is 4sqrt(130). 2.2 Determine the equation of diagonal AC in the form y = mx + c. In a rhombus, the diagonals are perpendicular to each other. The equation of diagonal BD is given as 3x - y = 0. Step 1: Find the gradient of diagonal BD. Rearrange the equation 3x - y = 0 to the form y = mx + c: y = 3x The gradient of BD, m_BD, is 3. Step 2: Find the gradient of diagonal AC. Since the diagonals are perpendicular, the product of their gradients is -1. m_AC × m_BD = -1 m_AC × 3 = -1 m_AC = -(1)/(3) Step 3: Use the point A (-8, 6) and the gradient m_AC = -(1)/(3) to find the equation of AC. Using the point-slope form y - y_1 = m(x - x_1): y - 6 = -(1)/(3)(x - (-8)) y - 6 = -(1)/(3)(x + 8) y - 6 = -(1)/(3)x - (8)/(3) Add 6 to both sides: y = -(1)/(3)x - (8)/(3) + 6 y = -(1)/(3)x - (8)/(3) + (18)/(3) y = -(1)/(3)x + (10)/(3) The equation of diagonal AC is y = -(1)/(3)x + (10)/(3). What's next?