DSQ: Calculating whether lines are perpendicular - GMAT Quantitative Reasoning
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The equations of two lines are:
Are these lines perpendicular?
Statement 1: 
Statement 2: 
The equations of two lines are:
Are these lines perpendicular?
Statement 1:
Statement 2:
The lines of the two equations must have slopes that are the opposites of each others reciprocals.
Write each equation in slope-intercept form:
As can be seen, knowing the value of 
is necessary and sufficient to answer the question. The value of 
is irrelevant.
The answer is that Statement 1 alone is sufficient to answer the question, but Statement 2 alone is not sufficient to answer the question.
The lines of the two equations must have slopes that are the opposites of each others reciprocals.
Write each equation in slope-intercept form:
As can be seen, knowing the value of
The answer is that Statement 1 alone is sufficient to answer the question, but Statement 2 alone is not sufficient to answer the question.
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Data Sufficiency Question
Is Line A perpendicular to the following line?
Statement 1: The slope of Line A is 3.
Statement 2: Line A passes through the point (2,3).
Data Sufficiency Question
Is Line A perpendicular to the following line?
Statement 1: The slope of Line A is 3.
Statement 2: Line A passes through the point (2,3).
To determine if two lines are perpendicular, only the slope needs to be considered. The slopes of perpendicular lines are the negative reciprocals of each other. Knowing a single point on the line is not sufficient, as an infinite number of lines can pass through and individual point.
To determine if two lines are perpendicular, only the slope needs to be considered. The slopes of perpendicular lines are the negative reciprocals of each other. Knowing a single point on the line is not sufficient, as an infinite number of lines can pass through and individual point.
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Refer to the above figure.
True or false: 
Statement 1: 
Statement 2: 
Refer to the above figure.
True or false:
Statement 1:
Statement 2:
Statement 1 alone establishes by definition that 
, but does not establish any relationship between 
and 
.
By Statement 2 alone, since alternating interior angles are congruent, 
, 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. By Statement 2, 
. 
and 
are corresponding angles formed by a transversal across parallel lines, so 
. 
is not a right angle, so 
.
Statement 1 alone establishes by definition that
By Statement 2 alone, since alternating interior angles are congruent,
Assume both statements are true. By Statement 2,
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Refer to the above figure. True or false: 
Statement 1: 
is equilateral.
Statement 2: Line 
bisects 
.
Refer to the above figure. True or false:
Statement 1:
Statement 2: Line
Statement 1 alone establishes nothing about the angle 
makes with 
, as it is not part of the triangle. Statement 2 alone only establishes that 
.
Assume both statements are true. Then 
is an altitude of an equilateral triangle, making it - and 
- perpendicular with the base 
- and 
.
Statement 1 alone establishes nothing about the angle
Assume both statements are true. Then
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Refer to the above figure. True or false: 
Statement 1: 
Statement 2: Line 
bisects 
.
Refer to the above figure. True or false:
Statement 1:
Statement 2: Line
Assume Statement 1 alone. Then, as a consequence of congruence, 
and 
are congruent. They form a linear pair of angles, so they are also supplementary. Two angles that are both congruent and supplementary must be right angles, so 
.
Assume Statement 2 alone. Then 
, but without any other information about the angles that 
or 
make with 
, it cannot be determined whether 
or not.
Assume Statement 1 alone. Then, as a consequence of congruence,
Assume Statement 2 alone. Then
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Statement 1:
Refer to the above figure. Are the lines perpendicular?
Statement 1: 
Statement 2: 
Statement 1:
Refer to the above figure. Are the lines perpendicular?
Statement 1:
Statement 2:
Assume Statement 1 alone. The measure of one of the angles formed is
degrees.
Assume Statement 2 alone.
By substituting 
for 
, one angle measure becomes
The marked angles are a linear pair and thus their angle measures add up to 180 degrees; therefore, we can set up an equation:
or 
yields illegal angle measures - for example,
yields angle measures 
for both angles; the angles are right and the lines are perpendicular.
Assume Statement 1 alone. The measure of one of the angles formed is
Assume Statement 2 alone.
By substituting
The marked angles are a linear pair and thus their angle measures add up to 180 degrees; therefore, we can set up an equation:
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Consider the lines of the equations
and
Are these two lines parallel, perpendicular, or neither?
Statement 1: 
Statement 2: 
Consider the lines of the equations
and
Are these two lines parallel, perpendicular, or neither?
Statement 1:
Statement 2:
Since the two equations are in slope-intercept form, 
coefficients 
and 
are the slopes of the two lines.
If 
, then this tells us that one of slopes 
and 
is positive and one is negative; this only eliminates the possibility of the lines being parallel.
If 
- or, equivalently, 
, then each of the slopes 
and 
is the opposite of the reciprocal of the other. This makes the lines perpendicular.
Since the two equations are in slope-intercept form,
If
If
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You are given two lines. Are they perpendicular?
Statement 1: The product of their slopes is 1.
Statement 2: The sum of their slopes is 
.
You are given two lines. Are they perpendicular?
Statement 1: The product of their slopes is 1.
Statement 2: The sum of their slopes is
The product of the slopes of two perpendicular lines is 
, so from Statement 1 alone, from the fact that this product is 1, you can deduce that the slopes are not perpendicular.
Statement 2 is neither necessary nor helpful, since the sum of the slopes is irrelevant to the question.
The product of the slopes of two perpendicular lines is
Statement 2 is neither necessary nor helpful, since the sum of the slopes is irrelevant to the question.
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Are Line 1 and Line 2 on the coordinate plane perpendicular?
Statement 1: Line 1 is the line of the equation 
.
Statement 2: Line 2 has no 
-intercept.
Are Line 1 and Line 2 on the coordinate plane perpendicular?
Statement 1: Line 1 is the line of the equation
Statement 2: Line 2 has no
We need to know the slopes of both lines to answer this question. Each statement gives information about only one line, so neither alone gives a definite answer.
Statement 1 tells us that Line 1 is vertical, since it is a line of the equation 
, for some real 
. Statement 2 tells us that the line, not crossing the 
-axis, must be parallel to the 
-axis and, subsequently, horizontal. A vertical line and a horizontal line are perpendicular, so the two statements together answer the question.
We need to know the slopes of both lines to answer this question. Each statement gives information about only one line, so neither alone gives a definite answer.
Statement 1 tells us that Line 1 is vertical, since it is a line of the equation
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Are linear equations 
and 
perpendicular?
I) 
pass through the points 
and 
.
II) 
passes through the point 
and has a 
-intercept of 
.
Are linear equations
I)
II)
To find whether these two functions are perpendicular we need to find each of their slopes.
Perpendicular lines have opposite, reciprocal slopes.
Use I) to find the slope of 
Use II) to find the slope of 
These are not opposite reciprocals, so 
and 
are not perpendicular.
To find whether these two functions are perpendicular we need to find each of their slopes.
Perpendicular lines have opposite, reciprocal slopes.
Use I) to find the slope of
Use II) to find the slope of
These are not opposite reciprocals, so
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