Coordinate Geometry - GMAT Quantitative
Card 0 of 1283
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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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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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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You are given two lines. Are they perpendicular?
Statement 1: The sum of their slopes is 
.
Statement 2: They have the same slope.
You are given two lines. Are they perpendicular?
Statement 1: The sum of their slopes is
Statement 2: They have the same slope.
Statement 2 alone tells us that the lines are parallel, not perpendicular. Statement 1 alone is neither necessary nor helpful, as the sum of the slopes is irrelevant.
Statement 2 alone tells us that the lines are parallel, not perpendicular. Statement 1 alone is neither necessary nor helpful, as the sum of the slopes is irrelevant.
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Consider this system of equations:
Does this system has exactly one solution?
Statement 1: The lines representing the equations are parallel.
Statement 2: 
Consider this system of equations:
Does this system has exactly one solution?
Statement 1: The lines representing the equations are parallel.
Statement 2:
The solution of a system of linear equations is the point at which their lines intersect; if they are parallel, then by definition, there is no such point, and the system has no solution.
If 
, then we can rewrite the second equation as
In slope-intercept form:
This line has a slope of 
. The other equation has a line with slope of 
also, as can be easily seen since it is already in slope-intercept form. Since both equations have lines with the same slope, they are either the same line or parallel lines; either way, the system does not have exactly one solution.
The solution of a system of linear equations is the point at which their lines intersect; if they are parallel, then by definition, there is no such point, and the system has no solution.
If
In slope-intercept form:
This line has a slope of
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Given 
, find the equation of 
, a line 
to 
.
I) 
.
II) The 
-intercept of 
is at 
.
Given
I)
II) The
To find the equation of a perpendicular line you need the slope of the line and a point on the line. We can find the slope by knowing g(x).
I) Gives us a point on h(x).
II) Gives us the y-intercept of h(x).
Either of these will be sufficient to find the rest of our equation.
To find the equation of a perpendicular line you need the slope of the line and a point on the line. We can find the slope by knowing g(x).
I) Gives us a point on h(x).
II) Gives us the y-intercept of h(x).
Either of these will be sufficient to find the rest of our equation.
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You are given two lines. Are they parallel?
Statement 1: The product of their slopes is 
.
Statement 2: One has positive slope; one has negative slope.
You are given two lines. Are they parallel?
Statement 1: The product of their slopes is
Statement 2: One has positive slope; one has negative slope.
Two parallel lines must have the same slope. Therefore, the product of the slopes will be the product of two real numbers of like sign, which must be positive. Each of the two statements contradicts this, so either statement alone answers the question.
Two parallel lines must have the same slope. Therefore, the product of the slopes will be the product of two real numbers of like sign, which must be positive. Each of the two statements contradicts this, so either statement alone answers the question.
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Line AB is perpindicular to Line BC. Find the equation for Line AB.
1. Point B (the intersection of these two lines) is (2,5).
2. Line BC is parallel to the line y=2x.
Line AB is perpindicular to Line BC. Find the equation for Line AB.
1. Point B (the intersection of these two lines) is (2,5).
2. Line BC is parallel to the line y=2x.
To find the equation of any line, we need 2 pieces of information, the slope of the line and any point on the line. From statement 1, we get a point on Line AB. From statement 2, we get the slope of Line BC. Since we know that AB is perpindicular to BC, we can derive the slope of AB from the slope of BC. Therefore to find the equation of the line, we need the information from both statements.
To find the equation of any line, we need 2 pieces of information, the slope of the line and any point on the line. From statement 1, we get a point on Line AB. From statement 2, we get the slope of Line BC. Since we know that AB is perpindicular to BC, we can derive the slope of AB from the slope of BC. Therefore to find the equation of the line, we need the information from both statements.
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Find the graph of the linear function 
.
I) 
passes through the points 
and 
.
II) 
intercepts the 
-axis at 
.
Find the graph of the linear function
I)
II)
Find the graph of the linear function 
.
I) 
passes through the points 
and 
.
II) 
intercepts the 
-axis at .
Using I), we can find the slope of the function, and then we can start at either point and extend the slope in either direction to find our graph:
So, using I) we are able to find the slope, from which we can find our graph
II) gives us one point, but without any more information, we cannot use II) by itself to find the rest of the graph
So:
Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.
Find the graph of the linear function
I)
II)
Using I), we can find the slope of the function, and then we can start at either point and extend the slope in either direction to find our graph:
So, using I) we are able to find the slope, from which we can find our graph
II) gives us one point, but without any more information, we cannot use II) by itself to find the rest of the graph
So:
Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.
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Find the graph of 
.
I) 
is a linear equation which passes through the point 
.
II) 
crosses the y-axis at 1300.
Find the graph of
I)
II)
Find the graph of 
.
I) 
is a linear equation which passes through the point 
.
II) 
crosses the y-axis at 1300.
To graph a linear equation, we need some combination of slope, y-intercept, or two points.
Statement I tells us 
is linear and gives us one point.
Statement II gives us the y-intercept of 
.
We can use Statement I and Statement II to find the slope of 
. Then, we can plot the given points and continue the line in either direction to get our graph.
Slope:
Plugging in the provided value of 
, 1300, we have the equation of the line 
:
Find the graph of
I)
II)
To graph a linear equation, we need some combination of slope, y-intercept, or two points.
Statement I tells us
Statement II gives us the y-intercept of
We can use Statement I and Statement II to find the slope of
Slope:
Plugging in the provided value of
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Graph a line, if possible.
Statement 1: The slope is 4.
Statement 2: The y-intercept is 4.
Graph a line, if possible.
Statement 1: The slope is 4.
Statement 2: The y-intercept is 4.
Statement 1): The slope is 4.
Write the slope-intercept form, and substitute the slope.
The point and the y-intercept are unknown. Either of these will be needed to solve for the graph of this line.
Statement 1) by itself is not sufficient to graph a line.
Statement 2): The y-intercept is 4.
Substitute the y-intercept into the incomplete formula.
The function 
can then be graphed on the x-y coordinate plane.
Therefore:
Statement 1): The slope is 4.
Write the slope-intercept form, and substitute the slope.
The point and the y-intercept are unknown. Either of these will be needed to solve for the graph of this line.
Statement 1) by itself is not sufficient to graph a line.
Statement 2): The y-intercept is 4.
Substitute the y-intercept into the incomplete formula.
The function
Therefore:
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Define a function 
as follows:
for nonzero real numbers 
.
Give the equation of the vertical asymptote of the graph of 
.
Statement 1: 
Statement 2: 
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
.
Statement 1 alone gives that 
. 
is the reciprocal of this, or 
, and 
, so the vertical asymptote is 
.
Statement 2 alone gives no clue about either 
, 
, or their relationship.
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
Statement 1 alone gives that
Statement 2 alone gives no clue about either
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Define a function 
as follows:
for nonzero real numbers 
.
Give the equation of the vertical asymptote of the graph of 
.
Statement 1: 
Statement 2: 
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Since a logarithm of a nonpositive number cannot be taken,
Therefore, the vertical asymptote must be the vertical line of the equation
.
Each of Statement 1 and Statement 2 gives us only one of 
and 
. However, the two together tell us that
making the vertical asymptote
.
Since a logarithm of a nonpositive number cannot be taken,
Therefore, the vertical asymptote must be the vertical line of the equation
Each of Statement 1 and Statement 2 gives us only one of
making the vertical asymptote
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Define a function 
as follows:
for nonzero real numbers 
.
Does the graph of 
have a 
-intercept?
Statement 1: 
.
Statement 2: 
.
Define a function
for nonzero real numbers
Does the graph of
Statement 1:
Statement 2:
The 
-intercept of the graph of the function 
, if there is one, occurs at the point with 
-coordinate 0. Therefore, we find 
:
This expression is defined if and only if 
is a positive value. Statement 1 gives 
as positive, so it follows that the graph indeed has a 
-intercept. Statement 2, which only gives 
, is irrelevant.
The
This expression is defined if and only if
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Define a function 
as follows:
for nonzero real numbers 
.
Does the graph of 
have a 
-intercept?
Statement 1: 
.
Statement 2: 
and 
have different signs.
Define a function
for nonzero real numbers
Does the graph of
Statement 1:
Statement 2:
The 
-intercept of the graph of the function 
, if there is one, occurs at the point with 
-coordinate 0. Therefore, we find 
:
This expression is defined if and only if 
is a positive value. However, the two statements together do not give this information; the values of 
and 
from Statement 1 are irrelevant, and Statement 2 does not reveal which of 
and 
is positive and which is negative.
The
This expression is defined if and only if
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Define a function 
as follows:
for nonzero real numbers 
.
Where is the vertical asymptote of the graph of 
in relation to the 
-axis - is it to the left of it, to the right of it, or on it?
Statement 1: 
and 
are both positive.
Statement 2: 
and 
are of opposite sign.
Define a function
for nonzero real numbers
Where is the vertical asymptote of the graph of
Statement 1:
Statement 2:
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
.
Statement 1 gives irrelevant information. But Statement 2 alone gives sufficient information; since 
and 
are of opposite sign, their quotient 
is negative, and 
is positive. This locates the vertical asymptote on the right side of the 
-axis.
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
Statement 1 gives irrelevant information. But Statement 2 alone gives sufficient information; since
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Define a function 
as follows:
for nonzero real numbers 
.
Where is the vertical asymptote of the graph of 
in relation to the 
-axis - is it to the left of it, to the right of it, or on it?
Statement 1: 
and 
are both positive.
Statement 2: 
and 
are of opposite sign.
Define a function
for nonzero real numbers
Where is the vertical asymptote of the graph of
Statement 1:
Statement 2:
Since only positive numbers have logarithms, the expression 
must be positive, so
Therefore, the vertical asymptote must be the vertical line of the equation
.
In order to determine which side of the 
-axis the vertical asymptote falls, it is necessary to find the sign of 
; if it is negative, it is on the left side, if it is positive, it is on the right side.
Assume both statements are true. By Statement 1, 
is positive. If 
is positive, then 
is negative, and vice versa. However, Statement 2, which mentions 
, does not give its actual sign - just the fact that its sign is the opposite of that of 
, which we are not given either. The two statements therefore give insufficient information.
Since only positive numbers have logarithms, the expression
Therefore, the vertical asymptote must be the vertical line of the equation
In order to determine which side of the
Assume both statements are true. By Statement 1,
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