Question

Find the angle between the x-axis and the line joining the points (3, -1) and (4, -2).

Answer 1

Hint: The slope of the line is the angle between the x-axis and the line. So, to find the angle between the two lines, it is sufficient to find the slope of the line formed by joining the two given points. Mathematically, the slope between two points (x1, y1) and (h, k) can be represented as-
${\text{m}} = \dfrac{{{{\text{y}}_2} - {{\text{y}}_1}}}{{{{\text{x}}_2} - {{\text{x}}_1}}}$

Complete step-by-step answer:
First we will find the slope of the line formed by (3, -1) and (4, -2). This can be done as-
${\text{m}} = \dfrac{{ - 2 - \left( { - 1} \right)}}{{4 - 3}} = - \dfrac{1}{1} = - 1$
The slope of the line is the angle between the x-axis and the line, so the tangent of the slope is the slope of the line itself. We know that,
${\text{m}} = tan\theta $
$ - 1 = tan\theta $
${\theta} = {\tan ^{ - 1}}\left( { - 1} \right)$
${\theta} = {135^{\text{o}}}$
This is the angle between the x-axis and the line joining the points (3, -1), (4, -2).

Note: Whenever we have to find the angle between the axes and any line, we assume that the measurement is taken from the positive x-axis in the anti-clockwise direction, unless mentioned otherwise. For example, if the angle was asked from the negative x-axis, then the answer would be ${45^{\text{o}}}$.