Inclination and slope of a line
Inclination θ is the angle a line makes with the positive x-axis. Slope m = tan θ, or (y₂ − y₁)/(x₂ − x₁) using two points.
Inclination and slope
The inclination of a line is the angle θ it makes with the positive x-axis, measured anticlockwise. The slope (or gradient) tells you how steep the line is. We write it as m. If θ is the inclination, then m = tan θ. Slope is also 'rise over run': the change in y divided by the change in x. So for two points, m = (y₂ − y₁)/(x₂ − x₁). If slope AB = slope BC, the lines AB and BC have the same direction and share point B, so A, B and C are collinear.
What the slope tells you
| Inclination θ | Slope m | The line |
|---|---|---|
| θ = 0° | m = 0 | Horizontal (parallel to the x-axis) |
| 0° < θ < 90° (acute) | Positive | Rises from left to right |
| θ = 90° | Not defined | Vertical (parallel to the y-axis) |
| 90° < θ < 180° (obtuse) | Negative | Falls from left to right |
Example 1: slope from two points
Find the slope of the line through A(−2, 3) and B(4, −6). Is its inclination acute or obtuse?
- Formula: m = (y₂ − y₁)/(x₂ − x₁)
- m = (−6 − 3)/(4 − (−2))
- m = −9/6 = −3/2
- The slope is negative, so the line falls from left to right.
Slope = −3/2. The inclination is obtuse.
Example 2: a playground slide
The top of a straight slide is 2 m above the ground. Its foot is 2√3 m away, measured along the ground from the point just below the top. Find the slope and the inclination of the slide.
- Rise = 2 m and run = 2√3 m
- m = rise/run = 2/(2√3) = 1/√3
- tan θ = 1/√3, and tan 30° = 1/√3
- So θ = 30°
Slope = 1/√3 and inclination = 30°.
Example 3: collinear points using slopes
Using slopes, show that A(−2, −3), B(3, 0) and C(8, 3) are collinear.
- Formula: m = (y₂ − y₁)/(x₂ − x₁)
- Slope of AB = (0 − (−3))/(3 − (−2)) = 3/5
- Slope of BC = (3 − 0)/(8 − 3) = 3/5
- Slope of AB = slope of BC, and B is common to both.
- So A, B and C lie on one straight line.
A, B and C are collinear.
Signboards on ghat roads, like the roads to Ooty or Kodaikanal, warn drivers about steep slopes. A ramp for wheelchairs must have a small slope, so it is safe to climb.
1 marks: “What is the slope of a line whose inclination is 60°?” m = tan 60° = **√3**.
2 marks: “Find the slope of the line joining (5, −1) and (−3, 7). Is its inclination acute or obtuse?” m = (7 − (−1))/(−3 − 5) = 8/(−8) = **−1**. The slope is negative, so the inclination is **obtuse**.
5 marks: “Using slopes, show that A(−2, −3), B(3, 0) and C(8, 3) are collinear. Find k if D(k, 6) also lies on this line.” Slope AB = 3/5 and slope BC = 3/5, with B common, so the points are collinear. Slope CD = (6 − 3)/(k − 8) = 3/5 gives k − 8 = 5, so **k = 13**.
Rise over run: uphill is +, downhill is −, flat is 0, straight up has no slope.