Slope of a Line

5.2.2 Slope of a Line

If a line has inclination $\theta$, then $\tan\theta$ — when it exists — is called the slope of the line, denoted $m$:

Activity. For any two points $A(x_1,y_1)$, $B(x_2,y_2)$ on a non-vertical line of inclination $\theta$, one can verify geometrically (by dropping a perpendicular from one point to the horizontal through the other, forming a right triangle whose legs are $y_2-y_1$ and $x_2-x_1$) that

Because the Y-axis (and any line parallel to it) has inclination $90^\circ$, where tangent is undefined, the slope of the Y-axis — and of any vertical line — is not defined. The slope of the X-axis, and of any line parallel to it, is $0$ (since $\tan 0^\circ = 0$).

Remark. Two lines are parallel if and only if they have the same slope (this follows immediately from the inclination remark above, since $\tan$ is one-to-one on $[0^\circ,180^\circ)$ apart from the vertical case).

Worked Example 1. Find the slope of a line whose inclination is $60^\circ$. Slope $= \tan 60^\circ = \sqrt{3}$. …