GMAT Quantitative › Calculating whether lines are perpendicular
Figure NOT drawn to scale.
Refer to the above figure.
True or false:
Statement 1: is a right angle.
Statement 2: and
are supplementary.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
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.
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
.
Refer to the above figure. . True or false:
Statement 1:
Statement 2: and
are supplementary.
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.
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.
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,
.
Which of the following choices give the slopes of two perpendicular lines?
undefined,
We can eliminate the choice immediately since the slopes of two perpendicular lines cannot have the same sign. We can also eliminate
and undefined,
, since a line with slope 0 and a line with undefined slope are perpendicular to each other, not a line of slope -1 or 1.
Of the two remaining choices, we check for the choice that includes two numbers whose product is -1.
and
so is the correct choice.
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,
What is the slope of any line that is perpendicular to ?
None of the answers provided
For a given line defined by the equation
with slope
, any line perpendicular to
has a slope of
, or the negative reciprocal of
. Since the slope of the provided line
in this instance is
, then the slope of any line perpendicular to
is
.
What is the slope of any line that is perpendicular to ?
None of the above
For a given line defined by the equation
with slope
, any line perpendicular to
has a slope of
, or the negative reciprocal of
. Since the slope of the provided line
in this instance is
, then the slope of any line perpendicular to
is
.
What is the slope of any line that is perpendicular to ?
None of the above
For a given line defined by the equation
with slope
, any line perpendicular to
has a slope of
, or the negative reciprocal of
. Since the slope of the provided line
in this instance is
, then the slope of any line perpendicular to
is
.
Which of the following lines is perpendicular to
Two of the answers are correct.
Given a line defined by the equation
with a slope of
, any line perpendicular to
would have a slope that is the negative reciprocal of
,
. Given our equation
, we know that
and that
.
There are two answer choices with this slope, and
.
A given line is defined by the equation
. Which of the following lines would be perpendicular to line
?
Not enough information provided
For any line with an equation
and slope
, a line that is perpendicular to
must have a slope of
, or the negative reciprocal of
. Given
, we know that
and therefore know that
.
Only one equation above has a slope of :
.
Determine whether the lines with equations and
are perpendicular.
They are not perpendicular
They are perpendicular
There is not enough information to determine the answer
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.