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 problems: a) (i) Find the coordinates of D Assuming ABCD is a parallelogram, the diagonals bisect each other. This means the midpoint of AC is the same as the midpoint of BD. Given points: A(1,2), B(3,-1), C(-1,-1). Let D(x_D, y_D). Step 1: Find the midpoint of AC. M_AC = ((x_A + x_C)/(2), (y_A + y_C)/(2)) M_AC = ((1 + (-1))/(2), (2 + (-1))/(2)) M_AC = ((0)/(2), (1)/(2)) M_AC = (0, (1)/(2)) Step 2: Set the midpoint of BD equal to M_AC and solve for x_D and y_D. M_BD = ((x_B + x_D)/(2), (y_B + y_D)/(2)) ((3 + x_D)/(2), (-1 + y_D)/(2)) = (0, (1)/(2)) Equating the x-coordinates: (3 + x_D)/(2) = 0 3 + x_D = 0 x_D = -3 Equating the y-coordinates: (-1 + y_D)/(2) = (1)/(2) -1 + y_D = 1 y_D = 1 + 1 y_D = 2 The coordinates of D are (-3, 2). b) Find the value of x in the diagram The diagram shows a triangle with an exterior angle at P and two remote interior angles at Q and R. According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Step 1: Set up the equation based on the Exterior Angle Theorem. The exterior angle at P is 3x - 18^. The remote interior angle at Q is 52^. The remote interior angle at R is 2x + 10^. 3x - 18 = 52 + (2x + 10) Step 2: Simplify the equation. 3x - 18 = 2x + 62 Step 3: Solve for x. 3x - 2x = 62 + 18 x = 80 The value of x is 80. c) Simplify ((3)/(5) + 1(1)/(2)) ÷ (1)/(4) × 3(1)/(3) Step 1: Convert mixed numbers to improper fractions. 1(1)/(2) = (1 × 2 + 1)/(2) = (3)/(2) 3(1)/(3) = (3 × 3 + 1)/(3) = (10)/(3) The expression becomes: ((3)/(5) + (3)/(2)) ÷ (1)/(4) × (10)/(3) Step 2: Perform the addition inside the parentheses. Find a common denominator for (3)/(5) and (3)/(2), which is 10. (3)/(5) + (3)/(2) = (3 × 2)/(5 × 2) + (3 × 5)/(2 × 5) = (6)/(10) + (15)/(10) = (21)/(10) The expression is now: (21)/(10) ÷ (1)/(4) × (10)/(3) Step 3: Perform division and multiplication from left to right. First, the division. To divide by a fraction, multiply by its reciprocal. (21)/(10) ÷ (1)/(4) = (21)/(10) × (4)/(1) = (84)/(10) = (42)/(5) The expression is now: (42)/(5) × (10)/(3) Step 4: Perform the multiplication. (42)/(5) × (10)/(3) = (42 × 10)/(5 × 3) = (420)/(15) Step 5: Simplify the fraction. (420)/(15) = (84)/(3) = 28 The simplified value is 28. d) (i) Make m the subject of the formula t = (a - m)/(1 + bm) Step 1: Multiply both sides by (1 + bm) to eliminate the denominator. t(1 + bm) = a - m Step 2: Distribute t on the left side. t + tbm = a - m Step 3: Gather all terms containing m on one side and terms without m on the other side. tbm + m = a - t Step 4: Factor out m from the terms on the left side. m(tb + 1) = a - t Step 5: Divide both sides by (tb + 1) to isolate m. m = (a - t)/(tb + 1) The formula with m as the subject is m = (a - t)/(tb + 1).