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.
ScanSolve AI Answer
Here are the solutions for the requested problems from the image. 1.a) Calculate the angle formed by the two lines. Step 1: Identify the coordinates of the points on the lines. Line 1 passes through A(0, 5) and C(4, 0). Line 2 passes through A(0, 5) and B(-3, 0). Step 2: Calculate the gradient of Line 1 (m_1). m_1 = (y_2 - y_1)/(x_2 - x_1) = (0 - 5)/(4 - 0) = (-5)/(4) Step 3: Calculate the gradient of Line 2 (m_2). m_2 = (y_2 - y_1)/(x_2 - x_1) = (0 - 5)/(-3 - 0) = (-5)/(-3) = (5)/(3) Step 4: Use the formula for the angle between two lines. = | (m_1 - m_2)/(1 + m_1 m_2) | = | (-5)/(4) - (5)/(3)1 + (-(5)/(4))((5)/(3)) | = | (-15 - 20)/(12)1 - (25)/(12) | = | (-35)/(12)(12 - 25)/(12) | = | (-35)/(12)(-13)/(12) | = | (-35)/(-13) | = (35)/(13) Step 5: Calculate and round to two decimal places. = ((35)/(13)) ≈ 69.593^ ≈ 69.59^ 1.c) Calculate the angle formed by the two lines. Step 1: Identify the coordinates of the points on the lines. Line 1 passes through A(-2, 1) and B(1, 5). Line 2 passes through C(0, 4) and D(3, 2). Step 2: Calculate the gradient of Line 1 (m_1). m_1 = (y_2 - y_1)/(x_2 - x_1) = (5 - 1)/(1 - (-2)) = (4)/(3) Step 3: Calculate the gradient of Line 2 (m_2). m_2 = (y_2 - y_1)/(x_2 - x_1) = (2 - 4)/(3 - 0) = (-2)/(3) Step 4: Use the formula for the angle between two lines. = | (m_1 - m_2)/(1 + m_1 m_2) | = | (4)/(3) - (-