Find the angle between the -axis and the line joining the points and .
The angle between the x-axis and the line joining the points (3, -1) and (4, -2) is 135°.
Explanation
This question tests the student's understanding of slope and its relationship with the angle a line makes with the x-axis. The slope formula m = (y₂ - y₁)/(x₂ - x₁) is fundamental in coordinate geometry. When the slope is known, the angle θ that the line makes with the positive direction of the x-axis can be found using the relation tan(θ) = m. Since the slope is negative (-1), the line makes an obtuse angle with the positive x-axis.
Solution Steps
Step 1: Find the slope of the line joining points (3, -1) and (4, -2). Using the formula m = (y₂ - y₁)/(x₂ - x₁) m = (-2 - (-1))/(4 - 3) = (-2 + 1)/1 = -1/1 = -1
Step 2: Find the angle between the line and the x-axis. The angle θ between a line and the x-axis is given by: tan(θ) = m = -1 θ = tan⁻¹(-1)
Step 3: Determine the angle in standard form. Since tan⁻¹(-1) = -45° (principal value), and we express angles as positive values measured counterclockwise from the positive x-axis: θ = 180° - 45° = 135°
Alternatively, the acute angle between the line and the x-axis is 45°.