GMAT Math : Calculating whether lines are perpendicular

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Example Question #1 : Calculating Whether Lines Are Perpendicular

Find the equation of the line that is perpendicular to the line connecting the points \dpi{100} \small (0,-4)\ and\ (-1,-7).

Possible Answers:

the line between points \dpi{100} \small (0,0)\ and\ (2,2)

the line between the points \dpi{100} \small (3,0)\ and\ (-3,2)

\dpi{100} \small y=3x-1

\dpi{100} \small y=-4x+8

\dpi{100} \small y=\frac{x}{3}+1

Correct answer:

the line between the points \dpi{100} \small (3,0)\ and\ (-3,2)

Explanation:

Lines are perpendicular if their slopes are negative reciprocals of each other. First we need to find the slope of the line in the question stem.

slope = \frac{rise}{run} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-7 + 4}{-1 - 0} = \frac{-3}{-1} = 3

The negative reciprocal of 3 is \dpi{100} \small -\frac{1}{3}, so our answer will have a slope of \dpi{100} \small -\frac{1}{3}. Let's go through the answer choices and see.

\dpi{100} \small y=3x-1: This line is of the form \dpi{100} \small y=mx+b, where \dpi{100} \small m is the slope. The slope is 3, so this line is parallel, not perpendicular, to our line in question.

\dpi{100} \small y=-4x+8: The slope here is \dpi{100} \small -4, also wrong.

\dpi{100} \small y=\frac{x}{3}+1: The slope of this line is \dpi{100} \small \frac{1}{3}. This is the reciprocal, but not the negative reciprocal, so this is also incorrect.

The line between the points \dpi{100} \small (3,0)\ and\ (-3,2):\dpi{100} \small slope = \frac{2}{(-3-3)}=\frac{2}{-6}=-\frac{1}{3}.

This is the correct answer! Let's check the last answer choice as well.

The line between points \dpi{100} \small (0,0)\ and\ (2,2):\dpi{100} \small slope = \frac{2}{2}=1, which is incorrect.

Example Question #2 : Calculating Whether Lines Are Perpendicular

Determine whether the lines with equations  and  are perpendicular.

Possible Answers:

They are perpendicular

There is not enough information to determine the answer

They are not perpendicular

Correct answer:

They are not perpendicular

Explanation:

If two equations are perpendicular, then they will have inverse negative slopes of each other.  So if we compare the slopes of the two equations, then we can find the answer.  For the first equation we have  

so the slope is 

So for the equations to be perpendicular, the other equation needs to have a slope of 3.  For the second equation, we have 

 

so the slope is 

Since the slope of the second equation is not equal to 3, then the lines are not perpendicular.

Example Question #3 : Calculating Whether Lines Are Perpendicular

Transversal

Refer to the above figure. . True or false: 

Statement 1: 

Statement 2:  and  are supplementary.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If transversal  crosses two parallel lines  and , then same-side interior angles are supplementary, so  and  are supplementary angles. Also, corresponding angles are congruent, so .

By Statement 1 alone, angles  and  are congruent as well as supplementary; by Statement 2 alone,  and  are also supplementary as well as congruent. Two angles that are both supplementary and congruent are both right angles, so from either statement alone,  and  intersect at right angles, so, consequently, .

Example Question #4 : Calculating Whether Lines Are Perpendicular

Transversal

Figure NOT drawn to scale.

Refer to the above figure.

True or false: 

Statement 1:  is a right angle.

Statement 2:  and  are supplementary.

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone establishes by definition that , but does not establish any relationship between  and .

By Statement 2 alone, since same-side interior angles are supplementary, , but no conclusion can be drawn about the relationship of , since the actual measures of the angles are not given.

Assume both statements are true. If two lines are parallel, then any line in their plane perpendicular to one must be perpendicular to the other.  and , so it can be established that .

Example Question #5 : Calculating Whether Lines Are Perpendicular

Find the equation of the line that is perpendicular to the following equation and passes through the point .

Possible Answers:

Correct answer:

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:

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 , so

becomes

.

If we flip , 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:

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, , and solve for :

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

This equation satisfies the conditions of being perpendicular to our initial equation and passing through .

Example Question #6 : Calculating Whether Lines Are Perpendicular

Which of the following lines is perpendicular to ?

Possible Answers:

Two of the equations are perpendicular to the given line.

Correct answer:

Explanation:

In order for a line  to be perpendicular to another line  defined by the equation  , 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 .

 

In this instance, , so . Therefore, the correct solution is .

Example Question #7 : Calculating Whether Lines Are Perpendicular

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

Possible Answers:

Not enough information provided

Correct answer:

Explanation:

In order for a line  to be perpendicular to another line  defined by the equation  , 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 .

Given that we have a line  with a slope , we can therefore conclude that any perpendicular line would have a slope .

Example Question #8 : Calculating Whether Lines Are Perpendicular

Which of the following lines are perpendicular to ?

Possible Answers:

Two answers are perpendicular to the given line.

Correct answer:

Two answers are perpendicular to the given line.

Explanation:

In order for a line  to be perpendicular to another line  defined by the equation  , 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 .

Since in this instance the slope . Two of the above answers have this as their slope, so therefore that is the answer to our question.

Example Question #9 : Calculating Whether Lines Are Perpendicular

Do the functions  and  intersect at a ninety-degree angle, and how can you tell?

Possible Answers:

No, because  and  never intersect.

No, because  and  have different slopes.

Yes, because  and  have the same y-intercept.

It is impossible to determine from the information provided.

Yes, because the slope of  is the reciprocal of the slope of  and it has the opposite sign.

Correct answer:

Yes, because the slope of  is the reciprocal of the slope of  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:

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.

Example Question #10 : Calculating Whether Lines Are Perpendicular

Find the slope of a line that is perpendicular to the line running through the points  and 

Possible Answers:

Not enough information provided.

Correct answer:

Explanation:

To find the slope  of the line running through  and , we use the following equation:

The slope of any line perpendicular to the given line would have a slope that is the negative reciprocal of , or . Therefore, 

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