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 for 3.2, 3.3, and 4: 3.2 Size of A Step 1: Calculate the gradient of AC. Given A(-2; 10) and C(4; -2). m_AC = (-2 - 10)/(4 - (-2)) = (-12)/(6) = -2 Step 2: Use the gradient of AB, m_AB = 7, from 1.3. Step 3: Use the formula for the angle between two lines with gradients m_1 and m_2: = | (m_1 - m_2)/(1 + m_1 m_2) |. Let m_1 = m_AB = 7 and m_2 = m_AC = -2. A = | (7 - (-2))/(1 + (7)(-2)) | A = | (9)/(1 - 14) | A = | (9)/(-13) | A = (9)/(13) Step 4: Calculate the angle A. A = ((9)/(13)) A ≈ 34.70^ The size of A is 34.70^. 3.3 Coordinates of the point of intersection of the diagonals of parallelogram ACES, where S is a point in the first quadrant. Step 1: The diagonals of a parallelogram bisect each other. The point of intersection is the midpoint of either diagonal. Step 2: Use the coordinates of A(-2; 10) and E(12; 0) to find the midpoint M of diagonal AE. M_x = (x_A + x_E)/(2) = (-2 + 12)/(2) = (10)/(2) = 5 M_y = (y_A + y_E)/(2) = (10 + 0)/(2) = (10)/(2) = 5 The coordinates of the point of intersection of the diagonals are (5; 5). 4 Calculate the coordinates of T. Step 1: Calculate the length of BE using B(-4; -4) and E(12; 0). BE = sqrt((12 - (-4))^2 + (0 - (-4))^2) BE = sqrt((16)^2 + (4)^2) BE = sqrt(256 + 16) BE = sqrt(272) BE = sqrt(16 × 17) = 4sqrt(17) units This confirms the given length. Step 2: Use the distance formula for ET with E(12; 0) and T(p; p), and set ET = 4sqrt(17). ET = sqrt((p - 12)^2 + (p - 0)^2) 4sqrt(17) = sqrt((p - 12)^2 + p^2) Step 3: Square both sides and solve for p. (4sqrt(17))^2 = (p - 12)^2 + p^2 16 × 17 = p^2 - 24p + 144 + p^2 272 = 2p^2 - 24p + 144 0 = 2p^2 - 24p + 144 - 272 0 = 2p^2 - 24p - 128 Divide by 2: 0 = p^2 - 12p - 64 Step 4: Factor the quadratic equation. (p - 16)(p + 4) = 0 This gives two possible values for p: p = 16 or p = -4. Step 5: Apply the condition p > 0. Since p > 0, we choose p = 16. The coordinates of T(p; p) are (16; 16). Last free one today — make it count tomorrow, or type /upgrade for unlimited.