## Perpendicular Lines

Perpendicular lines are two lines that intersect at a right angle (90 degrees). To find the equation of a perpendicular line, we need to know the equation of the other line and the slope of the perpendicular line.

## What Is Perpendicular Lines Formula?

The formula for perpendicular lines involves finding the slope of one line and then using that slope to determine the slope of a line that is perpendicular to it.

Given two lines with slopes m1 and m2, if they are perpendicular, then their slopes satisfy the following relationship:

**m1 x m2 = -1**

In other words, the product of the slopes of two perpendicular lines is always equal to -1. This formula allows us to determine the slope of a line that is perpendicular to another line, given the slope of the first line.

Once we know the slope of the perpendicular line, we can use the point-slope form of the equation of a line to find its equation. The point-slope form of the equation of a line is:

y – y1 = m(x – x1)

where m is the slope of the line and (x1, y1) is a point on the line.

### Examples Using Perpendicular Lines Formula

#### Example 1: Find the equation of the line perpendicular to y = 2x + 1 that passes through the point (3, 4).

First, we need to find the slope of the given line, which is 2. Then, using the perpendicular lines formula, we can find the slope of the line perpendicular to it:

2 x m = -1 m = -1/2

Now that we have the slope (-1/2) and a point (3, 4) on the line, we can use the point-slope form of the equation of a line to find its equation:

y – 4 = (-1/2)(x – 3) y = (-1/2)x + 7/2

So the equation of the line perpendicular to y = 2x + 1 that passes through (3, 4) is y = (-1/2)x + 7/2.

#### Example 2: Find the equation of the line perpendicular to 3x – 4y = 5 that passes through the point (2, -1).

First, we need to rearrange the equation into slope-intercept form (y = mx + b):

3x – 4y = 5 -4y = -3x + 5 y = (3/4)x – 5/4

Now we can use the perpendicular lines formula to find the slope of the line perpendicular to it:

(3/4) x m = -1 m = -4/3

Using the point-slope form of the equation of a line with the slope (-4/3) and the point (2, -1), we can find the equation of the perpendicular line:

y – (-1) = (-4/3)(x – 2) y = (-4/3)x + 2/3

So the equation of the line perpendicular to 3x – 4y = 5 that passes through (2, -1) is y = (-4/3)x + 2/3.

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