Calculating the equation of a perpendicular line - GMAT Quantitative Reasoning

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Question

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

Answer

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 $-\frac{1}{2}$ .

$m=-\frac{1}{2}$ and :

$y-y_{1}=m(x-x_{1})$

$y-1=-\frac{1}{2}(x-5)$

$y-1=-\frac{1}{2}x+\frac{5}{2}$

$y=-\frac{1}{2}x+\frac{5}{2}+1$

$y=-\frac{1}{2}x+\frac{5}{2}+\frac{2}{2}$

$y=-\frac{1}{2}x+\frac{7}{2}$

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Question

Write the equation of a line that is perpendicular to $y=\frac{-1}{2}x+4$ and goes through point ?

Answer

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

The given line, $y=\frac{-1}{2}x+4$ , has a slope of $-\frac{1}{2}$ , 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:

$y-y_{1}=m(x-x_{1})$

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Question

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

Answer

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:

$m_{perp}=-\frac{1}{m}=-\frac{1}{3}$

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, :

$4=-\frac{1}{3}(2)+b$

$b=\frac{14}{3}$

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

$y=-\frac{1}{3}x+\frac{14}{3}$

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Question

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

$\small h(x)=-\frac{5}3x-12$

Answer

Given

$\small h(x)=-\frac{5}3x-12$

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

Perpendicular lines have opposite reciprocal slopes.

So we get our slope to be

$\small \frac{3}{5}$

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

$0=14\cdot \frac{3}{5}+b$

$\small b=-\frac{42}{5}$ .

Making our answer

$\small y=\frac{3x}{5}-\frac{42}{5}$ .

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Question

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

Answer

Given a line defined by the equation with slope , any line that is perpendicular to must have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Since , the slope is and the slope of any line parallel to must have a slope of $-\frac{1}{m}=-\frac{1}{4}$ .

Since also needs to have a -intercept of , then the equation for must be $g(x)=-\frac{1}{4}x+7$ .

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Question

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

Answer

Given a line defined by the equation with slope , any line that is perpendicular to must have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Since , the slope is and the slope of any line parallel to must have a slope of $-\frac{1}{m}=-\frac{1}{-5}=\frac{1}{5}$ .

Since also needs to have a -intercept of , then the equation for must be $g(x)=\frac{1}{5}x+8$ .

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Question

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

Answer

Given a line defined by the equation with slope , any line that is perpendicular to must have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Since , the slope is and the slope of any line parallel to must have a slope of $-\frac{1}{m}=-\frac{1}{7}$ .

Since also needs to have a -intercept of , then the equation for must be $g(x)=-\frac{1}{7}x-9$ .

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