# Lesson Explainer: Equations of Parallel and Perpendicular Lines Mathematics

In this explainer, we will learn how to write the equation of a line parallel or perpendicular to another line.

Parallel lines are straight lines that never intersect. To understand the link between parallel lines and their slopes, let us consider two lines of equations and

We can solve for the point of intersection by setting the expressions for to be equal:

Now, we rearrange so that the -terms are on the same side of the equation:

By factoring, we find

Finally, we note that we can isolate the variable by dividing by , and so, only if , this gives

This shows us that two lines intersect only if their slopes are not equal. Conversely, we can conclude that lines of the same slope do not intersect; a pair of parallel lines will have the same slopes.

We can see this by considering the slopes of a pair of parallel lines.

For every units we go across on either line, we must travel units in the vertical direction on both lines; otherwise, the lines will intersect. It is worth noting that there is one small problem with this reasoning, which is if we have vertical lines. In this case, we cannot talk about the slope, since vertical lines do not have a slope. However, we can note that a pair of distinct vertical lines will be parallel. We have the following result.

### Property: Slopes of Parallel Lines

If two nonvertical lines are parallel, then they have the same slopes.

If two distinct lines have the same slopes or are both vertical, then they are parallel.

This allows us to check if two lines are parallel. For example, consider the lines and . We can recall that a line given in the form has a slope of and a -intercept of . The first line is given in this form, so its slope is given by the coefficient of , which is . We can subtract from both sides of the equation of the second line to get

The coefficient of is , so its slope is also . It is important to check that the -intercepts of the two lines are different. If they had the same -intercept, then the two lines would actually be the same line (called coincident lines).

Since the two lines are distinct and have the same slopes, we can conclude that they are parallel. This property allows us to check if any two lines are parallel.

We can now ask the question of how to check if two lines are perpendicular. We can do this by sketching any nonvertical lines, including a right triangle representing its slope.

A line perpendicular to this line will meet the line at right angles. We can find any line like this by rotating the red line . If we rotate the line , we will also rotate the triangle .

We can then calculate the slope of the perpendicular line: for every units we move across, we move units down. Its slope is . Thus, if a line has slope , then the lines perpendicular to this line have slope . We can think of this in two ways: either we take the negative of the reciprocal of the slope or the product of the slopes as . The same is true in reverse: if two lines have slopes whose product is , then they are perpendicular. We have the following property.

### Property: Slopes of Perpendicular Lines

If two nonvertical lines are perpendicular, then their slopes are the negative of the reciprocal of each other. Alternatively, the product of their slopes gives .

If two lines have slopes and such that , then they are perpendicular.

The only caveat to this property is if we have horizontal or vertical lines. In this case, we can note that horizontal and vertical lines are perpendicular to each other even though their slopes do not satisfy this property.

In our first example, we will determine the relationship between a pair of lines from their equations.

### Example 1: Identifying Whether Two Lines Are Parallel, Perpendicular, or Otherwise

How would you describe the relation between the lines and ?

1. Parallel
2. Coincident
3. Perpendicular
4. Intersecting and not perpendicular

To determine if the lines are parallel or perpendicular, we first want to find their slopes. We can do this by writing both equations in the form , where is the slope and is the -intercept. The first equation is already in this form, so it has slope . We can rearrange the second equation to get

Therefore, the second line also has a slope of . Since both lines have the same slopes, they are either parallel or coincident. We can see that the first line has a -intercept of 9 but the second line has a -intercept of . Since the two lines pass through different -intercepts, they cannot be the same line, so they are not coincident.

Hence, since the lines have the same slopes and are not coincident, they must be parallel which is option A.

In our next example, we will find the equation of a line given a point on the line and the equation of a parallel line.

### Example 2: Finding the Equation of a Line given a Point on the Line and a Parallel Line

Write, in the form , the equation of the line through that is parallel to the line .

We first recall that the equation of a line with a slope of that passes through can be written as . As we have the coordinates of a point on the line, we now need to find its slope. To do this, we use the fact that the line is parallel to the line . As parallel lines have the same slopes (unless they are both vertical), we want to find the slope of the line . We can do this by rewriting its equation in the form .

We add to both sides of the equation to get

The coefficient of is , so the slope of this line, and thus of our line, is .

Substituting , , and into the equation gives us

We can now expand the brackets and rearrange to get

In our next example, we will find the equation of a line given a point on the line and two points on another parallel line.

### Example 3: Finding the Equation of a Line given a Point on the Line and Two Points on Another Parallel Line

Find, in slopeβintercept form, the equation of the straight line passing through the point and parallel to the straight line passing through the two points and .

We first recall that the slopeβintercept form of a line is the equation , where the line has a slope of and a -intercept of . We are not given the slope or -intercept of this line.

Instead, we are given a point on the line and two points on a parallel line. We can recall that for the lines to be parallel, they need to have the same slopes. We can determine the slope of a line through and by using the formula .

Substituting , , , and into the formula for the slope yields

Therefore, the slope of our line is .

Now, we can write the equation of our line in the form , where is its slope and are the coordinates of a point it passes through. Substituting , , and into this equation gives us

The last step is to rearrange this equation into the form . We start by expanding the brackets to get

We now add 1 to both sides of the equation to obtain

In our next example, we will find the equation of a line given a point on the line and the equation of a perpendicular line.

### Example 4: Finding the Equation of a Line given a Point on the Line and a Perpendicular Line

Find, in slopeβintercept form, the equation of the line perpendicular to that passes through the point .

We first recall that the slopeβintercept form of a line is the equation , where the line has a slope of and a -intercept of . We are not given the slope or -intercept of this line.

Instead, we are given a point on the line and the equation of a perpendicular line. We can determine the slope of the line by noting that it is perpendicular to the line ; perpendicular lines have slopes that multiply to give (unless one is a vertical line).

We can see that we are given the equation of the perpendicular line in slopeβintercept form. The coefficient of is 2, so the slope of this line is 2. The slope of the line we want to find is the negative of the reciprocal of this value. We have

We know that the equation of a line with a slope of that passes through can be written in the form .

Substituting , , and into this equation gives us

We can now expand the brackets and rearrange to get

In our next example, we will find the equation of a line given a point on the line and two points on a perpendicular line.

### Example 5: Finding the Equation of a Line given a Point on the Line and Two Points on Another Perpendicular Line

Find the equation of the straight line passing through the point and perpendicular to the straight line passing through the points and .

We want to determine the equation of a straight line given a point on the line and two points on a perpendicular line. We can do this by recalling that the equation of a line of slope that passes through the point is . We already know that our line passes through the point , so we just need its slope.

We can find the slope of the line by recalling that its product with the slope of the line perpendicular to it will be . We can determine the slope of a line passing through and using the formula . Substituting , , , and into this formula gives us the slope of the perpendicular line:

Taking the negative of the reciprocal of this value gives us the slope of our line:

We can now substitute , , and into the equation of a line to get

We can now expand the brackets and rearrange to obtain

In our next example, we will determine whether the lines between two pairs of points are parallel, perpendicular, or neither.

### Example 6: Determining Whether the Lines Between Given Points Are Parallel, Perpendicular, or Neither

Given that the coordinates of the points , , , and are , , , and , respectively, determine whether and are parallel, perpendicular, or neither.

We can check the relationship between a pair of lines by comparing the slopes. We recall that parallel lines have the same slopes and perpendicular lines have slopes that multiply to give , provided that neither line is vertical.

We can calculate the slope of a line passing through and using the formula . Substituting , , , and into the formula gives

Substituting , , , and into the formula gives

We see that the product of the slopes of the lines is :

Thus, the lines are perpendicular.

In our final example, we will use the fact that adjacent sides in a rectangle are perpendicular and opposite sides are parallel to determine the coordinates of the final vertex in a rectangle given the coordinates of the other three vertices.

### Example 7: Determining Whether the Lines Between Given Points Are Parallel, Perpendicular, or Neither

The vertices of a rectangle have the coordinates , , , and . Determine the values of and .

We first recall that the adjacent sides in a rectangle are perpendicular. This means that is perpendicular to . We can find expressions for the slopes of these lines and then use the fact that the lines are parallel to find an equation involving and .

We can calculate the slope of a line passing through and using the formula . Substituting , , , and into the formula gives

Substituting , , , and into the formula gives

Since the lines are parallel, they will have the same slopes. Hence,

We can rearrange this equation to get

This is not enough information to find the values of and . We can follow this process again with and . This time, the sides are adjacent, so they must be perpendicular.

Substituting , , , and into the formula gives

We can note that is neither horizontal nor vertical, so will not be horizontal or vertical. Thus, the product of the slopes of the lines will be , since they are perpendicular:

We can rearrange this equation to get

We now have a pair of simultaneous equations involving and . We can solve these to determine the values of and . Using the second equation, we have

We can substitute this expression for into the equation to get

Expanding the brackets yields

We can then solve for

Substituting into the equation gives us

Hence, and .

Let us finish by recapping some of the important points from this explainer.

### Key Points

• If two nonvertical lines are parallel, then they have the same slopes.
• If two distinct lines have the same slopes or are both vertical, then they are parallel.
• If two nonvertical lines are perpendicular, then their slopes are the negatives of the reciprocals of each other. Alternatively, the product of their slopes gives .
• If two lines have slopes and such that , then they are perpendicular.
• Distinct vertical lines are parallel to each other.
• Horizontal and vertical lines are perpendicular to each other.
• We can check if the sides of polygons are parallel or perpendicular from the coordinates of their vertices by comparing the slopes of the sides.