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
Step 1: Determine the gradient of PR (Question 1.1). The coordinates of P are (9; 2) and R are (-4; -4). The formula for the gradient m between two points (x_1, y_1) and (x_2, y_2) is: m = (y_2 - y_1)/(x_2 - x_1) Let (x_1, y_1) = R(-4, -4) and (x_2, y_2) = P(9, 2). m_PR = (2 - (-4))/(9 - (-4)) m_PR = (2 + 4)/(9 + 4) m_PR = (6)/(13) The gradient of PR is (6)/(13). Step 2: Find the angle of inclination of PR with the positive x-axis. Let be the angle of inclination of line PR with the positive x-axis. We know that m_PR = . = (6)/(13) = ((6)/(13)) ≈ 24.77^ Step 3: Relate to and explain why 90^ - is used. The angle is defined as the angle between the y-axis and the line PR. The x-axis and y-axis are perpendicular, meaning they form a 90^ angle. If is the angle the line PR makes with the positive x-axis, then the acute angle it makes with the y-axis is the complement of . Therefore, = 90^ - . This is because the sum of the angles in the right-angled triangle formed by the line PR, the x-axis, and a vertical line from a point on PR to the x-axis, or by the line PR, the y-axis, and a horizontal line from a point on PR to the y-axis, must be 180^. Since the angle between the axes is 90^, the other two acute angles must sum to 90^. Step 4: Calculate the size of (Question 1.2). = 90^ - = 90^ - 24.77^ ≈ 65.23^ Rounding to one decimal place: ≈ 65.2^ The size of is 65.2^. What's next?