Algebra Help: How to Tell If Two Lines Are Parallel, Perpendicular or Neither from Coordinates

In Algebra a common problem you will be asked to solve is whether two lines are parallel, perpendicular, or neither by looking at the coordinates.

How to Find the Slope of a Line
To solve this problem you need to understand how to find the slope of a line from the coordinates.
Assume we have line AB, where A=(-1,1) and B=(0,5). In this coordinate notation the first value is the X value (horizontal) and the second value is the Y value (vertical).

The formula for the slope of a line =( Y2 – Y1)/(X2 – X1)

The slope of line AB, therefore, would be (5-1)/(0+1) = 4

How to Tell if Two Lines are Parallel
When we look at two lines, two lines are parallel if they have the same slope.

So, if we have line CD, where C=(2,-2) and D=(2,3), the slope of line CD would be (2+2)/(3-2)= 4. This tell us that line AB and CD are parallel, because they have the same slope.

Negative Reciprocals and How to Tell if Two Lines are Perpendicular

Two lines that are perpendicular have slopes that are negative reciprocals of each other. To find the reciprocal of an integer simply put 1 on top and divide by the integer. For example, the reciprocal of 4 is 1/4. To find the reciprocal of a fraction simply flip the denominator (bottom) and the numerator (top). For example, the reciprocal of 4/5 is 5/4. To make a reciprocal negative simply multiply it by negative 1. The negative reciprocal of -3, therefore, would be 1/3.

Looking at our two parallel lines AB and CD, in order for line EF to be perpendicular to these lines, it would have to have a slope of -1/4.

How to Tell if Two Lines Intersect
The only time two lines DO NOT intersect are when they are parallel (have the same slope). So, if after finding the slope of your two lines you see that the slopes are NOT the same, you know that they will eventually intersect.


John H. Saxon, Jr. Algebra 1

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