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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You're on a roll — here's the solution for question 12: 12. The points A(1, 3), B(-2, 5), C(1, 7) and D(x, y) are the vertices of the parallelogram ABCD. (i) Find AB, BC and AC; Step 1: Calculate vector AB. To find the vector from point P(x_1, y_1) to Q(x_2, y_2), we use PQ = x_2 - x_1 \\ y_2 - y_1 . For AB, A(1, 3) and B(-2, 5): AB = -2 - 1 \\ 5 - 3 = -3 \\ 2 AB = -3 \\ 2 Step 2: Calculate vector BC. For BC, B(-2, 5) and C(1, 7): BC = 1 - (-2) \\ 7 - 5 = 1 + 2 \\ 2 = 3 \\ 2 BC = 3 \\ 2 Step 3: Calculate vector AC. For AC, A(1, 3) and C(1, 7): AC = 1 - 1 \\ 7 - 3 = 0 \\ 4 AC = 0 \\ 4 (ii) the coordinates of D. Step 1: Use the property of a parallelogram that opposite sides are parallel and equal in length. Therefore, their vectors are equal. For parallelogram ABCD, AD = BC. Let D = (x, y). AD = x - 1 \\ y - 3 From part (i), we know BC = 3 \\ 2 . Step 2: Equate the components of the vectors AD and BC to find x and y. x - 1 \\ y - 3 = 3 \\ 2 Equating the x-components: x - 1 = 3 x = 3 + 1 x = 4 Equating the y-components: y - 3 = 2 y = 2 + 3 y = 5 The coordinates of D are (4, 5). D(4, 5) What's next? Send 'em!