# GMAT Math : Calculating the equation of a perpendicular line

## Example Questions

### Example Question #1 : Perpendicular Lines

What is the equation of the line that is perpendicular to and goes through point ?

Explanation:

Perpendicular lines have slopes that are negative reciprocals of each other.

The slope for the given line is , from  , where  is the slope. Therefore, the negative reciprocal is .

and      :

### Example Question #1 : Calculating The Equation Of A Perpendicular Line

Write the equation of a line that is perpendicular to  and goes through point ?

Explanation:

A perpendicular line has a negative reciprocal slope to the given line.

The given line, , has a slope of , as is the slope in the standard form equation  .

Slope of perpendicular line:

Point:

Using the point slope formula, we can solve for the equation:

### Example Question #3 : Perpendicular Lines

Given , find the equation of a line that is perpendicular to  and goes through the point .

Explanation:

Given

We need a perpendicular line going through (14,0).

Perpendicular lines have opposite reciprocal slopes.

So we get our slope to be

Next, plug in all our knowns into  and solve for .

.

.

### Example Question #4 : Perpendicular Lines

Given the function , which of the following is the equation of a line perpendicular to  and has a -intercept of ?

Explanation:

Given a line  defined by the equation  with slope , any line that is perpendicular to  must have a slope, or the negative reciprocal of .

Since , the slope  is  and the slope of any line  parallel to  must have a slope of .

Since  also needs to have a -intercept of , then the equation for  must be .

### Example Question #5 : Perpendicular Lines

Given the function , which of the following is the equation of a line perpendicular to  and has a -intercept of ?

Explanation:

Given a line  defined by the equation  with slope , any line that is perpendicular to  must have a slope, or the negative reciprocal of .

Since , the slope  is  and the slope of any line  parallel to  must have a slope of .

Since  also needs to have a -intercept of , then the equation for  must be .

### Example Question #6 : Perpendicular Lines

Given the function , which of the following is the equation of a line perpendicular to  and has a -intercept of ?

None of the above

Explanation:

Given a line  defined by the equation  with slope , any line that is perpendicular to  must have a slope, or the negative reciprocal of .

Since , the slope  is  and the slope of any line  parallel to  must have a slope of .

Since  also needs to have a -intercept of , then the equation for  must be .

### Example Question #7 : Perpendicular Lines

Determine the equation of a line perpendicular to  at the point .

Explanation:

The equation of a line in standard form is written as follows:

Where  is the slope of the line and  is the y intercept. First, we can determine the slope of the perpendicular line using the knowledge that its slope must be the negative reciprocal of the slope of the line to which it is perpendicular. For the given line, we can see that , so the slope of a line perpendicular to it will be the negative reciprocal of that value, which gives us:

Now that we know the slope of the perpendicular line, we can plug its value into the formula for a line along with the coordinates of the given point, allowing us to calculate the -intercept, :

We now have the slope and the -intercept of the perpendicular line, which is all we need to write its equation in standard form:

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