$m \,=\, \tan{\theta}$

The tangent of inclination of a straight line is called the slope of straight line.

$\overleftrightarrow{PQ}$ is a straight line in Cartesian coordinate system with some inclination.

Draw a parallel line to $x$-axis from point $P$ and also draw a perpendicular line towards $x$-axis from point $Q$. The both lines are intersected at point $R$ perpendicularly and it formed a right triangle ($\Delta RPQ$) geometrically.

The lengths of $\overline{QR}$ and $\overline{PR}$ represent vertical rise and horizontal distance of points of $P$ and $Q$ of the straight line. The slope of this straight line is ratio of vertical rise to horizontal distance of points of the straight line.

The slope of straight line $\overleftrightarrow{PQ}$ is represented by letter $m$ mathematically in geometric system.

$m \,=\, \dfrac{Vertical \, Rise}{Horizontal \, Distance}$

$\implies m \,=\, \dfrac{QR}{PR}$

$\Delta RPQ$ is a right triangle and, $\overline{QR}$ and $\overline{PR}$ are opposite side and adjacent side of the right triangle. Theta ($(\theta)$) is inclination of the straight line and it is also angle of the right triangle. The ratio of $QR$ to $PR$ is tan of angle theta as per trigonometry.

$\,\,\, \therefore \,\,\,\,\,\, m \,=\, \tan{\theta}$

The derivation has proved that the slope of a straight line is equal to tangent of the inclination of the straight line.

Latest Math Topics

Latest Math Problems

Email subscription

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.