What is the perpendicular slope of

Perpendicular lines have slopes that are opposite reciprocals of each other. To find the slope of a line that is perpendicular to a given equation, find the opposite reciprocal of that slope. Check out this tutorial to learn how!

Perpendicular lines have slopes that are opposite reciprocals of on another. The slope of the line y = -3x + 1 is m = -3. Therefore, we take the opposite reciprocal of 3 to find the line perpendicular to the line y = -3x + 1 must have slope

When you graph two or more linear equations in a coordinate plane, they generally cross at a point. However, when two lines in a coordinate plane never cross, they are called parallel linesTwo or more lines that lie in the same plane but which never intersect.. You will also look at the case where two lines in a coordinate plane cross at a right angle. These are called perpendicular linesTwo lines that lie in the same plane and intersect at a `90^@` angle. . The slopes of the graphs in each of these cases have a special relationship to each other.

Parallel lines are two or more lines in a plane that never intersect. Examples of parallel lines are all around us, such as the opposite sides of a rectangular picture frame and the shelves of a bookcase.

Perpendicular lines are two or more lines that intersect at a `90`-degree angle, like the two lines drawn on this graph. These `90`-degree angles are also known as right angles.

Perpendicular lines are also everywhere, not just on graph paper but also in the world around us, from the crossing pattern of roads at an intersection to the colored lines of a plaid shirt.

Supplemental Interactive Activity

Explore lines in the interactive diagram below.

• Click and drag the dot in the “Equation” slider to choose one of five example equations. The equation will be graphed in blue.
• Then, click and drag the dot on the red line to make the line parallel or perpendicular to the blue line. (Be sure to move your cursor slowly.) When the lines are parallel or perpendicular, text will appear to let you know you’ve done it!
• Look at the slopes of the two parallel lines. What do you notice? Look at the slopes of two perpendicular lines. What do you notice?
• Choose another equation and try again.
• As you try more equations, look for the relationship between the slopes of parallel lines, and the slopes for perpendicular lines. For the last equation you try, can you predict what the slopes of the parallel and perpendicular lines should be?

Note the following:

Parallel Lines

Two non-vertical lines in a plane are parallel if they have both:

• the same slope
• different `y`-intercepts

Any two vertical lines in a plane are parallel.

Example

Problem

Find the slope of a line parallel to the line `y` = `-3x + 4`.

The given line is written in `y = mx + b` form, with `m = -3` and `b = 4`. The slope is `-3`.

Identify the slope of the given line.

Answer

The slope of the parallel line is `-3`.

A line parallel to the given line has the same slope.

Example

Problem

Determine whether the lines `y = 6x + 5` and `y = 6x - 1` are parallel.

The given lines are written in `y = mx + b` form, with `m = 6` for the first line and `m = 6` for the second line. The slope of both lines is `6`.

Identify the slopes of the given lines.

The first line has a `y`-intercept at `(0, 5)`, and the second line has a `y`-intercept at `(0, -1)`. They are not the same line.

Look at `b`, the `y`-value of the `y`-intercept, to see if the lines are perhaps exactly the same line, in which case we don’t say they are parallel.

Answer

The lines are parallel.

The slopes of the lines are the same and they have different `y`-intercepts, so they are not the same line and they are parallel.

Perpendicular Lines

Two non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other. If the slope of the first equation is `4`, then the slope of the second equation will need to be `-1/4` for the lines to be perpendicular.

You can also check the two slopes to see if the lines are perpendicular by multiplying the two slopes together. If they are perpendicular, the product of the slopes will be `-1`. For example, `4*-1/4=4/1*-1/4=-1`.

Example

Problem

Find the slope of a line perpendicular to the line `y = 2x - 6`.

The given line is written in `y = mx + b` form, with `m = 2` and `b = -6`. The slope is `2`.

Identify the slope of the given line.

Answer

The slope of the perpendicular line is `-1/2`.

To find the slope of a perpendicular line, find the reciprocal, `1/2`, and then find the opposite of this reciprocal `-1/2`.

Note that the product `2(-1/2)=2/1(-1/2)=-1`, so this means the slopes are perpendicular.

In the case where one of the lines is vertical, the slope of that line is undefined and it is not possible to calculate the product with an undefined number. When one line is vertical, the line perpendicular to it will be horizontal, having a slope of zero `(m = 0)`.

Example

Problem

Determine whether the lines `y=-8x + 5` and `y=1/8x-1` are parallel, perpendicular, or neither.

The given lines are written in `y = mx + b` form, with `m = -8` for the first line and `m=1/8` for the second line.

Identify the slopes of the given lines.

`-8!=1/8`, so the lines are not parallel.

The opposite reciprocal of `-8` is `1/8`, so the lines are perpendicular.

Determine if the slopes are the same or if they are opposite reciprocals.

Answer

The lines are perpendicular.

The slopes of the lines are opposite reciprocals, so the lines are perpendicular.

Which of the following lines are perpendicular to the line `y=7x+2/3`?

`y=-1/7x`

`y=-1/7x-12`

`y=1/7x-3/2`

`y=-7x+3/2`

A) `y=-1/7x text( and ) y=-1/7x-12`

B) `y=-1/7x-12 text ( and ) y=1/7x-3/2`

C) `y=-7x+3/2`

D) All of the lines are perpendicular.

A) `y=-1/7x text ( and ) y=-1/7x-12`

Correct. These lines both have a slope of `-1/7`, which is the opposite reciprocal of the slope of `7` in the original equation. Both of these lines are perpendicular to the original line.

B) `y=-1/7x-12 text( and ) y=1/7x-3/2`

Incorrect. Perpendicular lines have slopes that are opposite of the reciprocal of the other. Only `-1/7` is the opposite of the reciprocal of `7`. The correct answer is `y=-1/7x` and `y=-1/7x-12.`

C) `y=-7x+3/2`

Incorrect. Perpendicular lines have slopes that are the opposite of the reciprocal of the other. `-7` is the opposite of `7`, but it is not the opposite of the reciprocal of `7`. The correct answer is `y=-1/7x` and `y=-1/7x-12.`

D) All of the lines are perpendicular.

Incorrect. Only the lines that have a slope of `-1/7` , which is the opposite of the reciprocal of `7`, can be perpendicular to the original equation. The correct answer is `y=-1/7x` and `y=-1/7x-12.`

The relationships between slopes of parallel and perpendicular lines can be used to write equations of parallel and perpendicular lines.

Let’s start with an example involving parallel lines.

Example

Problem

Write the equation of a line that is parallel to the line `x - y = 5`  and goes through the point `(-2, 1)`.

Rewrite the line you want to be parallel to into the `y = mx + b` form, if needed.

In the equation above, `m = 1` and `b = -5`.

Since `m = 1`, the slope is `1`.

Identify the slope of the given line.

The slope of the parallel line is `1`.

To find the slope of a parallel line, use the same slope.

`y = mx + b`

`1 = 1(-2) + b`

Use the method for writing an equation from the slope and a point on the line. Substitute `1` for `m`, and the point `(-2, 1)` for `x` and `y`.

`1 = -2 + b`

`3 = b`

Solve for `b`.

Answer

`y = x + 3`

Write the equation using the new slope for `m` and the `b` you just found.

When you are working with perpendicular lines, you will usually be given one of the lines and an additional point.

Example

Problem

Write the equation of a line that contains the point `(1, 5)` and is perpendicular to the line `y = 2x - 6`.

The given line is written in `y = mx + b` form, with `m = 2` and `b = -6`. The slope is `2`.

Identify the slope of the line you want to be perpendicular to.

The slope of the perpendicular line is `-1/2`.

To find the slope of a perpendicular line, find the reciprocal, `1/2`, then the opposite, `-1/2`.

`y=mx+b`

`5=-1/2(1)+b`

Use the method for writing an equation from the slope and a point on the line. Substitute `-1/2` for `m`, and the point `(1, 5)` for `x` and `y`.

`5=-1/2+b`

`11/2=b`

Solve for `b`.

Answer

`y=-1/2x+11/2`

Write the equation using the new slope for `m` and the `b` you just found.

Which of the following is the equation of the line that is parallel to `y = -2x - 14` and goes through the point `(-3, 1)`?

A) `y=-2x+1`

B) `y=1/2x+2 1/2`

C) `y=1/2x+1`

D) `y=-2x-5`

A) `y=-2x+1`

Incorrect. Checking `(-3, 1)` into the equation gives `1 = -2(-3) + 1`, which is `1 = 6 + 1`. Since `6 + 1 = 7`, not `1`, this line cannot go through the point `(-3, 1)`. The correct answer is `y = -2x - 5`.

B) `y=1/2x+2 1/2`

Incorrect. Parallel lines have the same slope, so this equation should also have a slope of `-2`. The correct answer is `y = -2x - 5`.

C) `y=1/2x+1`

Incorrect. Parallel lines have the same slope, so this equation should also have a slope of `-2`. The correct answer is `y = -2x - 5`.

D) `y=-2x-5`

Correct. The line has the same slope as the original, so they are parallel. Checking `(-3, 1)` into the equation gives `1 = -2(-3) - 5`, or `1 = 6 - 5`, which is true. The line is parallel and goes through the point `(-3, 1)`.

Example

Problem

Write the equation of a line that is parallel to the line `y = 4.`

`y = 4`

`y = 0x + 4`

Rewrite the line into `y = mx + b` form, if needed. You may notice without doing this that `y = 4` is a horizontal line `4` units above the `x`-axis. Because it is horizontal, you know its slope is zero.

In the equation above, `m = 0` and `b = 4`.

Since `m = 0`, the slope is `0`. This is a horizontal line.

Identify the slope of the given line.

The slope of the parallel line is also `0`.

To find the slope of a parallel line, use the same slope.

`y = 10`

Since the parallel line will be a horizontal line, its form is `y = a` constant.

Pick a constant to find a parallel line.

Answer

`y = 10`

This line is parallel to `y = 4` and intersects the `y`-axis at `(0, 10)`.

When lines in a plane are parallel (that is, they never cross), they have the same slope. When lines are perpendicular (that is, they cross at a `90^@` angle), their slopes are opposite reciprocals of each other. The product of their slopes will be `-1`, except in the case where one of the lines is vertical, causing its slope to be undefined. You can use these relationships to find an equation of a line that goes through a particular point and is parallel or perpendicular to another line.

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