Co-ordinate Geometry
287. Slope & Inclination · m = tan θ
Slope equals tan(theta) for any valid position of C.
The slope (gradient) m of a line equals the tangent of its inclination: m = tan θ. A steeper line has a larger inclination and therefore a larger slope; a horizontal line (θ = 0°) has slope 0.
What this lesson covers
Try to break it
Drag C around. The line OC tilts, the inclination θ changes, and the slope m = tan(θ) tracks it. Drag C close to vertical and θ approaches 90° — the slope shoots toward infinity. Lay C horizontally and θ approaches 0° — the slope drops to 0. Try to read a finite slope when θ = 90°; impossible — that's why vertical lines have "undefined" slope.
How you build it
Draw a line and find its slope from its inclination.
- Point tool: plot A on the x-axis — this is the vertex of the inclination angle θ.
- Point tool: plot C up and to the right of A so the rise equals the run — a 45° inclination, slope m = tan 45° = 1.
- Line tool: click A, then C. Its slope m = rise / run = tan θ — for θ = 45°, m = 1.
The proof, step by step
Prove that the slope of a line equals the tangent of its inclination.
- Slope is defined as the ratio of the vertical change (Rise) to the horizontal change (Run).
- In the right-angled triangle formed by the line, Rise/Run equals tan(θ).
- Therefore, Slope m = tan θ.
Worked example
If the inclination of a line is 60°, find its slope.
Slope m = tan θ = tan 60° = √3.
- 1/2
- 1/√3
- √3 — correct
- 1