GMAT Math : Geometry

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Example Questions

Example Question #3 : Lines

Find the equation of the line that is perpendicular to the following equation and passes through the point (4,7) .

$y=-\frac{x}{2}+14$

Possible Answers:

y=2x+1

y=2x-14

y=4x-8

y=2x-1

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

Correct answer:

y=2x-1

Explanation:

To solve this equation, we want to begin by recalling how to find the slope of a perpendicular line. In this case, our original line is modeled by the following:

$y=-\frac{x}{2}+14$

To find the slope of any line perpendicular to the above equation, we simply need to take the reciprocal of the first slope, and then change its sign. Our original slope is $-\frac{1}{2}$ , so

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

becomes

{m}'=2 .

If we flip $\frac{1}{2}$ , we get , and the opposite sign of a negative is a positive; hence, our slope is positive .

So, we know our perpendicular line should look something like this:

y=2x+b

However, we need to find out what (our -intercept) is in order to complete our equation. To do so, we need to plug in the ordered pair we received in the question, (4,7) , and solve for :

7=2*4+b

7=8+b

7-8=b

b=-1

So, by putting everything together, we get our final equation:

y=2x-1

This equation satisfies the conditions of being perpendicular to our initial equation and passing through (4,7) .

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Example Question #2 : Lines

Which of the following lines is perpendicular to $y=-\frac{3}{4}x+7$ ?

Possible Answers:

$y=\frac{3}{4}x+7$

$y=-\frac{4}{3}x+7$

$y=-\frac{3}{4}x-7$

Two of the equations are perpendicular to the given line.

$y=\frac{4}{3}x+7$

Correct answer:

$y=\frac{4}{3}x+7$

Explanation:

In order for a line to be perpendicular to another line defined by the equation y=mx+b , the slope of line must be a negative reciprocal of the slope of line . Since line 's slope is in the slope-intercept equation above, line 's slope would therefore be $-\frac{1}{m}$ .

In this instance, $m=-\frac{3}{4}$ , so $-\frac{1}{m}=-\frac{1}{-\frac{3}{4}}=\frac{4}{3}$ . Therefore, the correct solution is $y=\frac{4}{3}x+7$ .

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Example Question #2 : Calculating Whether Lines Are Perpendicular

A given line has a slope of m=6 . What is the slope of any line perpendicular to ?

Possible Answers:

Not enough information provided

$-\frac{1}{6}$

Correct answer:

$-\frac{1}{6}$

Explanation:

Given that we have a line with a slope m=6 , we can therefore conclude that any perpendicular line would have a slope $-\frac{1}{m}=-\frac{1}{6}$ .

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Example Question #3 : Calculating Whether Lines Are Perpendicular

Which of the following lines are perpendicular to $y=\frac{5}{6}x+10$ ?

Possible Answers:

$y=-\frac{6}{5}x-7$

Two answers are perpendicular to the given line.

$y=-\frac{5}{6}x-2$

$y=\frac{5}{6}x-7$

$y=-\frac{6}{5}x+10$

Correct answer:

Two answers are perpendicular to the given line.

Explanation:

Since in this instance the slope $m=\frac{5}{6}$ , $-\frac{1}{m}=-\frac{1}{\frac{5}{6}}=-\frac{6}{5}$ . Two of the above answers have this as their slope, so therefore that is the answer to our question.

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Example Question #4 : Calculating Whether Lines Are Perpendicular

Do the functions f(x) and g(x) intersect at a ninety-degree angle, and how can you tell?

$\small f(x)=3x+7$

$\small g(x)=-\frac{1}{3}x+7$

Possible Answers:

It is impossible to determine from the information provided.

Yes, because f(x) and g(x) have the same y-intercept.

No, because f(x) and g(x) never intersect.

Yes, because the slope of g(x) is the reciprocal of the slope of f(x) and it has the opposite sign.

No, because f(x) and g(x) have different slopes.

Correct answer:

Yes, because the slope of g(x) is the reciprocal of the slope of f(x) and it has the opposite sign.

Explanation:

If two lines intersect at a ninety-degree angle, they are said to be perpendicular. Two lines are perpendicular if their slopes are opposite reciprocals. In this case:

$f(x)\:slope=\frac{3}{1}$

$g(x)\:slope=-\frac{1}{3}$

The two lines' slopes are reciprocals with opposing signs, so the answer is yes. Of our two yes answers, only one has the right explanation. Eliminate the option dealing with -intercepts.

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Example Question #5 : Calculating Whether Lines Are Perpendicular

Find the slope of a line that is perpendicular to the line running through the points (7,4) and (5,8) .

Possible Answers:

$\frac{1}{2}$

$-\frac{1}{2}$

Not enough information provided.

Correct answer:

$\frac{1}{2}$

Explanation:

To find the slope of the line running through (7,4) and (5,8) , we use the following equation:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{8-4}{5-7}=\frac{4}{-2}=-2$

The slope of any line perpendicular to the given line would have a slope that is the negative reciprocal of , or $-\frac{1}{m}$ . Therefore, $-\frac{1}{-2}=\frac{1}{2}$

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Example Question #11 : Lines

Which of the following lines is perpendicular to $y=\frac{3}{4}x+16$ ?

Possible Answers:

$y=-\frac{4}{3}x+12$

$y=\frac{3}{4}x+7$

$y=\frac{4}{3}x+8$

Not enough information provided.

$y=\frac{3}{4}x-6$

Correct answer:

$y=-\frac{4}{3}x+12$

Explanation:

Given a line defined by the equation y=mx+b with a slope of , any line perpendicular to would have a slope that is the negative reciprocal of , $-\frac{1}{m}$ . Given our equation $y=\frac{3}{4}x+16$ , we know that $m=\frac{3}{4}$ and that $-\frac{1}{m}=-\frac{1}{\frac{3}{4}}=-\frac{4}{3}$ .

The only answer choice with this slope is $y=-\frac{4}{3}x+12$ .

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Example Question #12 : Lines

Which of the following lines is perpendicular to y=5x-12

Possible Answers:

$y=-\frac{1}{5}x+7$

$y=-\frac{1}{5}x-4$

$y=\frac{1}{5}x-6$

Two of the answers are correct.

$y=\frac{1}{5}x+19$

Correct answer:

Two of the answers are correct.

Explanation:

Given a line defined by the equation y=mx+b with a slope of , any line perpendicular to would have a slope that is the negative reciprocal of , $-\frac{1}{m}$ . Given our equation y=5x-12 , we know that m=5 and that $-\frac{1}{m}=-\frac{1}{5}$ .

There are two answer choices with this slope, $y=-\frac{1}{5}x-4$ and $y=-\frac{1}{5}x+7$ .

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Example Question #13 : Lines

A given line is defined by the equation y=2x+5 . Which of the following lines would be perpendicular to line ?

Possible Answers:

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

y=2x+14

y=-2x+7

Not enough information provided

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

Correct answer:

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

Explanation:

For any line with an equation y=mx+b and slope , a line that is perpendicular to must have a slope of $-\frac{1}{m}$ , or the negative reciprocal of . Given y=2x+5 , we know that m=2 and therefore know that $-\frac{1}{m}=-\frac{1}{2}$ .

Only one equation above has a slope of $-\frac{1}{2}$ : $y=-\frac{1}{2}x+7$ .

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Example Question #14 : Lines

What is the slope of a line that is perpendicular to $y=-\frac{1}{7}x+9$

Possible Answers:

$\frac{1}{9}$

$-\frac{1}{9}$

Correct answer:

Explanation:

For any line with an equation y=mx+b and slope , a line that is perpendicular to must have a slope of $-\frac{1}{m}$ , or the negative reciprocal of . Given the equation $y=-\frac{1}{7}x+9$ , we know that $m=-\frac{1}{7}$ and therefore know that $-\frac{1}{m}=-\frac{1}{-\frac{1}{7}}=7$ .

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #3 : Lines

Example Question #2 : Lines

Example Question #2 : Calculating Whether Lines Are Perpendicular

Example Question #3 : Calculating Whether Lines Are Perpendicular

Example Question #4 : Calculating Whether Lines Are Perpendicular

Example Question #5 : Calculating Whether Lines Are Perpendicular

Example Question #11 : Lines

Example Question #12 : Lines

Example Question #13 : Lines

Example Question #14 : Lines

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