What is special about slopes of parallel and perpendicular lines?

2:54 AM

In coordinate geometry, parallel and perpendicular lines have special significance. First let’s recap what we mean by parallel and perpendicular lines and then see some important properties of these lines.

Parallel Lines

Parallel lines are lines that are on the same plane and never intersect. On an XY plane, parallel lines have the following properties:

Parallel lines have an equal distance between them e.g. in the following figure there is a unique distance " " between the two parallel lines
Parallel lines have the same slope in the XY plane e.g. in the following example, if the parallel lines are and , then

In coordinate geometry, two lines are parallel if their slopes are equal i.e. lines

and

are parallel if

Perpendicular Lines

Two lines are perpendicular if they intersect and form a right angle at the point of intersection.

In the coordinate plane, two lines are perpendicular if the product of their slopes are -ve reciprocal

Let’s solve a coordinate geometry problem based on what we just learnt:

Example:

Line

passes through the points point M (2,4) and point N (3,8). If a line

is drawn perpendicular to line

, what is the slope of

Solution:

For this question you need to find the slope of the line

that is perpendicular to line

. Since the two lines are perpendicular, we know that where

and

are slopes of line

and line

. Using the information in the question, let’s calculate the slope of line

using the coordinates of point M and N.

Slope of line s1 = (4-8) / (2-3) = 4

(As you can see, the order in which you list the points really doesn't matter, as long as you subtract the x-values in the same order as you subtracted the y-values. Because of this, the slope formula can be written as it is above, or alternatively it can be written as: