GMAT Math : Perpendicular Lines

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Example Question #1 : Perpendicular Lines

What is the equation of the line that is perpendicular to y=2x+10 and goes through point (5,1) ?

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

The slope for the given line is , from y=2x+10 (y=mx+b) , where is the slope. Therefore, the negative reciprocal is $-\frac{1}{2}$ .

$m=-\frac{1}{2}$ and (5,1) :

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

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

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

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

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

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

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Example Question #1 : Perpendicular Lines

Write the equation of a line that is perpendicular to $y=\frac{-1}{2}x+4$ and goes through point (0,6) ?

Possible Answers:

$y=\frac{-1}{2}x+6$

y=-2x+6

$y=\frac{1}{2}x+6$

y=2x+6

Correct answer:

y=2x+6

Explanation:

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 (y=mx+b) .

Slope of perpendicular line: m=2

Point: (0,6)

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

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

y-6=2(x-0)

y-6=2x

y=2x+6

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Example Question #2 : Perpendicular Lines

Given h(x) , find the equation of a line that is perpendicular to h(x) and goes through the point (14,0) .

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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 y=mx+b 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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Example Question #892 : Problem Solving Questions

Given the function f(x)=4x+9 , which of the following is the equation of a line perpendicular to f(x) and has a -intercept of ?

Possible Answers:

$g(x)=\frac{1}{4}x-7$

$g(x)=-\frac{1}{4}x+7$

g(x)=4x+9

$g(x)=-\frac{1}{4}x-7$

$g(x)=\frac{1}{4}x+7$

Correct answer:

$g(x)=-\frac{1}{4}x+7$

Explanation:

Given a line defined by the equation f(x)=mx+b with slope , any line that is perpendicular to must have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Since f(x)=4x+9 , the slope is and the slope of any line g(x) parallel to f(x) must have a slope of $-\frac{1}{m}=-\frac{1}{4}$ .

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

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Example Question #893 : Problem Solving Questions

Given the function f(x)=-5x-4 , which of the following is the equation of a line perpendicular to f(x) and has a -intercept of ?

Possible Answers:

$g(x)=\frac{1}{5}x-8$

$g(x)=\frac{1}{5}x+8$

g(x)=5x+8

$g(x)=-\frac{1}{5}x-8$

$g(x)=-\frac{1}{5}x+8$

Correct answer:

$g(x)=\frac{1}{5}x+8$

Explanation:

Given a line defined by the equation f(x)=mx+b with slope , any line that is perpendicular to must have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Since f(x)=-5x-4 , the slope is and the slope of any line g(x) parallel to f(x) must have a slope of $-\frac{1}{m}=-\frac{1}{-5}=\frac{1}{5}$ .

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

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Example Question #894 : Problem Solving Questions

Given the function f(x)=7x+2 , which of the following is the equation of a line perpendicular to f(x) and has a -intercept of ?

Possible Answers:

$g(x)=\frac{1}{7}x+9$

None of the above

$g(x)=-\frac{1}{7}x+9$

$g(x)=\frac{1}{7}x-9$

$g(x)=-\frac{1}{7}x-9$

Correct answer:

$g(x)=-\frac{1}{7}x-9$

Explanation:

Given a line defined by the equation f(x)=mx+b with slope , any line that is perpendicular to must have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Since f(x)=7x+2 , the slope is and the slope of any line g(x) parallel to f(x) must have a slope of $-\frac{1}{m}=-\frac{1}{7}$ .

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

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Example Question #895 : Problem Solving Questions

Determine the equation of a line perpendicular to y=3x-2 at the point (2,4) .

Possible Answers:

$y=-\frac{1}{3}x-\frac{10}{3}$

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

$y=\frac{1}{3}x+\frac{4}{3}$

$y=\frac{1}{3}x+4$

$y=-\frac{1}{3}x+\frac{7}{3}$

Correct answer:

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

Explanation:

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

y=mx+b

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 m=3 , 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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Example Question #1 : Calculating The Slope Of A Perpendicular Line

What is the slope of the line perpendicular to -9x-9y=9 ?

Possible Answers:

m=9

$m=-\frac{1}{9}$

m=1

m=-1

Correct answer:

m=1

Explanation:

Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, rewrite the equation in slope intercept form (y=mx+b) :

-9x-9y=9

-9y=9x+9

y=-x-1

Slope of given line:

Negative reciprocal: m=1

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

Line 1 is the line of the equation x + y = 17 . Line 2 is perpendicular to this line. What is the slope of Line 2?

Possible Answers:

$- \frac{1}{17}$

Correct answer:

Explanation:

Rewrite in slope-intercept form:

x + y = 17

x + y -x = 17-x

y = -x + 17

y = -1x + 17

The slope of the line is the coefficient of , which is . A line perpendicular to this has as its slope the opposite of the reciprocal of :

$m = -\frac{1}{-1} = 1$

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Example Question #2 : Calculating The Slope Of A Perpendicular Line

Given:

$\small f(x)=\frac{5}{6}x-17$

Calculate the slope of g(x) , a line perpendicular to f(x) .

Possible Answers:

$\small -\frac{5}{6}$

$\small -6$

$\small \frac{6}{5}$

$\small 5$

$\small -\frac{6}{5}$

Correct answer:

$\small -\frac{6}{5}$

Explanation:

To find the slope of a line perpendicular to a given line, simply take the opposite reciprocal of the slope of the given line.

Since f(x) is given in slope intercept form,

$y=mx+b \rightarrow m=slope$ .

Therefore our original slope is

$m=\small \frac{5}{6}$

So our new slope becomes:

$m_{new}=-\frac{1}{m}=-\frac{1}{\frac{5}{6}}=\small -\frac{6}{5}$

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GMAT Math : Perpendicular Lines

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Perpendicular Lines

Example Question #1 : Perpendicular Lines

Example Question #2 : Perpendicular Lines

Example Question #892 : Problem Solving Questions

Example Question #893 : Problem Solving Questions

Example Question #894 : Problem Solving Questions

Example Question #895 : Problem Solving Questions

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

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

Example Question #2 : Calculating The Slope Of A Perpendicular Line

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