All GMAT Math Resources
Example Questions
Example Question #621 : Gmat Quantitative Reasoning
Find the equation of the line that is perpendicular to the line connecting the points .
the line between the points
the line between points
the line between the points
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.
The negative reciprocal of 3 is , so our answer will have a slope of . Let's go through the answer choices and see.
: This line is of the form , where is the slope. The slope is 3, so this line is parallel, not perpendicular, to our line in question.
: The slope here is , also wrong.
: The slope of this line is . This is the reciprocal, but not the negative reciprocal, so this is also incorrect.
The line between the points :.
This is the correct answer! Let's check the last answer choice as well.
The line between points :, which is incorrect.
Example Question #1 : Lines
Determine whether the lines with equations and are perpendicular.
They are not perpendicular
There is not enough information to determine the answer
They are perpendicular
They are not perpendicular
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 #2 : Lines
Refer to the above figure. . True or false:
Statement 1:
Statement 2: and are supplementary.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT 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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
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 #623 : Gmat Quantitative Reasoning
Figure NOT drawn to scale.
Refer to the above figure.
True or false:
Statement 1: is a right angle.
Statement 2: and are supplementary.
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.
EITHER statement ALONE is 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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
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 #3 : Lines
Find the equation of the line that is perpendicular to the following equation and passes through the point .
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 #2 : Lines
Which of the following lines is perpendicular to ?
Two of the equations are perpendicular to the given line.
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 #2 : Calculating Whether Lines Are Perpendicular
A given line has a slope of . What is the slope of any line perpendicular to ?
Not enough information provided
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 #3 : Calculating Whether Lines Are Perpendicular
Which of the following lines are perpendicular to ?
Two answers are perpendicular to the given line.
Two answers are perpendicular to the given line.
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 #4 : Calculating Whether Lines Are Perpendicular
Do the functions and intersect at a ninety-degree angle, and how can you tell?
It is impossible to determine from the information provided.
Yes, because and have the same y-intercept.
No, because and never intersect.
Yes, because the slope of is the reciprocal of the slope of and it has the opposite sign.
No, because and have different slopes.
Yes, because the slope of is the reciprocal of the slope of and it has the opposite sign.
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 #5 : Calculating Whether Lines Are Perpendicular
Find the slope of a line that is perpendicular to the line running through the points and .
Not enough information provided.
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,