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 the given problems: The vertices of the figure are A(2; 6), B(-6; -4), and C(10; -6). 1.4.1 Determine the length of AC Step 1: Use the distance formula D = sqrt((x_2-x_1)^2 + (y_2-y_1)^2). For points A(2, 6) and C(10, -6): AC = sqrt((10-2)^2 + (-6-6)^2) Step 2: Calculate the differences in coordinates and square them. AC = sqrt((8)^2 + (-12)^2) AC = sqrt(64 + 144) Step 3: Add the squared values and find the square root. AC = sqrt(208) AC = sqrt(16 × 13) AC = 4sqrt(13) The length of AC is 4sqrt(13) units. 1.4.2 The midpoint of BC Step 1: Use the midpoint formula M = ((x_1+x_2)/(2), (y_1+y_2)/(2)). For points B(-6, -4) and C(10, -6): M_BC = ((-6+10)/(2), (-4+(-6))/(2)) Step 2: Calculate the sums and divide by 2. M_BC = ((4)/(2), (-10)/(2)) M_BC = (2, -5) The midpoint of BC is (2, -5). 1.4.3 The angle of inclination of AB Step 1: Calculate the gradient (slope) of AB using the formula m = (y_2-y_1)/(x_2-x_1). For points A(2, 6) and B(-6, -4): m_AB = (-4-6)/(-6-2) m_AB = (-10)/(-8) m_AB = (5)/(4) Step 2: Use the formula for the angle of inclination = (m). = (5)/(4) = ((5)/(4)) Step 3: Calculate the angle. ≈ 51.34^ The angle of inclination of AB is approximately 51.34^. Send me the next one 📸