Coordinate Geometry - GMAT Quantitative
Card 1 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
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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:
Tap to reveal answer
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).
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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:
Tap to reveal answer
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
Tap to reveal answer
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:
Tap to reveal answer
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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Given 
, find the equation of 
, a line 
to 
.
I) 
.
II) The 
-intercept of 
is at 
.
Given
I)
II) The
Tap to reveal answer
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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Ava was challenged by her teacher to fill in the triangle, the square, and the circle in the diagram below with three numbers to form the equation of a line with slope 1.
Did Ava succeed?
Statement 1: Ava wrote a 1 in the circle.
Statement 2: Ava wrote the same positive number in both the triangle and the square.
Ava was challenged by her teacher to fill in the triangle, the square, and the circle in the diagram below with three numbers to form the equation of a line with slope 1.
Did Ava succeed?
Statement 1: Ava wrote a 1 in the circle.
Statement 2: Ava wrote the same positive number in both the triangle and the square.
Tap to reveal answer
Replacing the shapes with variables, the template becomes
Divide by 
to get the equation in slope-intercept form:
The slope is 
.
The slope is the ratio of the number in the square to the number in the triangle, so the number in the circle is irrelevant, making Statement 1 unhelpful.
Assume Statement 2 alone. Since the numbers in the square and the triangle are equal, 
, and 
. Ava succeeded.
Replacing the shapes with variables, the template becomes
Divide by
The slope is
The slope is the ratio of the number in the square to the number in the triangle, so the number in the circle is irrelevant, making Statement 1 unhelpful.
Assume Statement 2 alone. Since the numbers in the square and the triangle are equal,
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John's teacher gave him two equations, each with one coefficient missing, as follows:
John was challenged to write one number in each shape in order to form two equations whose lines have the same slope. Did he succeed?
Statement 1: The number John wrote in the box is three times the number he wrote in the circle.
Statement 2: John wrote 
in the circle.
John's teacher gave him two equations, each with one coefficient missing, as follows:
John was challenged to write one number in each shape in order to form two equations whose lines have the same slope. Did he succeed?
Statement 1: The number John wrote in the box is three times the number he wrote in the circle.
Statement 2: John wrote
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Rewrite the two equations with variables rather than shapes:.
The first equation can be rewritten in slope-intercept form:
Its line has slope is 
, so if the number in the square is known, the slope is known.
is already in slope-intercept form; its line has slope 
, the number in the circle.
Statement 2 alone gives the number in the circle but provides no clue to the number in the square.
Now assume Statement 1 alone. Then 
. The slope of the first line is
,
the slope of the second line. Statement 1 provides sufficient proof that John was successful.
Rewrite the two equations with variables rather than shapes:.
The first equation can be rewritten in slope-intercept form:
Its line has slope is
is already in slope-intercept form; its line has slope
Statement 2 alone gives the number in the circle but provides no clue to the number in the square.
Now assume Statement 1 alone. Then
the slope of the second line. Statement 1 provides sufficient proof that John was successful.
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Line m is perpendicular to the line l which is defined by the equation 
. What is the value of 
?
(1) Line m passes through the point 
.
(2) Line l passes through the point 
.
Line m is perpendicular to the line l which is defined by the equation
(1) Line m passes through the point
(2) Line l passes through the point
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Statement 1 allows you to define the equation of line m, but does not provide enough information to solve for 
. There are still 3 variables 
and only two different equations to solve.
if 
, statement 2 supplies enough information to solve for b by substitution if 
is on the line.
Statement 1 allows you to define the equation of line m, but does not provide enough information to solve for
if
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Consider linear functions 
and 
.
I) 
at the point 
.
II) 
Is the point 
on the line 
?
Consider linear functions
I)
II)
Is the point
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Consider linear functions h(t) and g(t).
I) 
at the point 
II) 
Is the point 
on the line h(t)?
We can use II) and I) to find the slope of h(t)
Recall that perpendicular lines have opposite reciprocal slope. Thus, the slope of h(t) must be 
Next, we know that h(t) must pass through (6,4), so lets us that to find the y-intercept:
Next, check if (10,4) is on h(t) by plugging it in.
So, the point is not on the line, and we needed both statements to know.
Consider linear functions h(t) and g(t).
I)
II)
Is the point
We can use II) and I) to find the slope of h(t)
Recall that perpendicular lines have opposite reciprocal slope. Thus, the slope of h(t) must be
Next, we know that h(t) must pass through (6,4), so lets us that to find the y-intercept:
Next, check if (10,4) is on h(t) by plugging it in.
So, the point is not on the line, and we needed both statements to know.
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Veronica's teacher gave her two equations, the first with the coefficients of both variables missing, as follows:
Veronica was challenged to write one number in each shape in order to form an equation whose line has the same slope as that of the second equation. The only restriction was that she could not write a 5 in the square or a 3 in the circle.
Did Veronica write an equation with the correct slope?
Statement 1: Veronica wrote a negative integer in the square and a positive integer in the circle.
Statement 2: Veronica wrote an 8 in the circle.
Veronica's teacher gave her two equations, the first with the coefficients of both variables missing, as follows:
Veronica was challenged to write one number in each shape in order to form an equation whose line has the same slope as that of the second equation. The only restriction was that she could not write a 5 in the square or a 3 in the circle.
Did Veronica write an equation with the correct slope?
Statement 1: Veronica wrote a negative integer in the square and a positive integer in the circle.
Statement 2: Veronica wrote an 8 in the circle.
Tap to reveal answer
The slope of the line of
can be found by writing this equation in slope-intercept form:
The slope of the line is the coefficient of 
is 
, so Veronica must place the numbers in the shapes to yield an equation whose slope has this equation.
Rewrite the top equation as
The slope, in terms of 
and 
, can be found similarly:
Its slope is 
.
Statement 1 asserts that 
and 
are of unlike sign, so the slope 
must be negative. It cannot have sign 
, so the question is answered.
Assume Statement 2 alone. Then in the above equation, 
, so the slope is 
. The slope now depends on the value of 
, so Statement 2 gives insufficient information.
The slope of the line of
can be found by writing this equation in slope-intercept form:
The slope of the line is the coefficient of
Rewrite the top equation as
The slope, in terms of
Its slope is
Statement 1 asserts that
Assume Statement 2 alone. Then in the above equation,
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Give the slope of a line on the coordinate plane.
Statement 1: The line passes through the graph of the equation 
on the 
-axis.
Statement 2: The line passes through the graph of the equation 
on the 
-axis.
Give the slope of a line on the coordinate plane.
Statement 1: The line passes through the graph of the equation
Statement 2: The line passes through the graph of the equation
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The 
-intercept(s) of the graph of 
can be found by setting 
and solving for 
:
The graph has exactly one 
-intercept, 
.
The 
-intercept(s) of the graph of 
can be found by setting 
and solving for 
:
The graph has exactly one 
-intercept, 
.
In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. Each of the two statements yields one of the points, so neither statement alone is sufficient to determine the slope. The two statements together, however, yield two points, and are therefore enough to determine the slope.
The
The graph has exactly one
The
The graph has exactly one
In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. Each of the two statements yields one of the points, so neither statement alone is sufficient to determine the slope. The two statements together, however, yield two points, and are therefore enough to determine the slope.
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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.
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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)
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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)
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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.
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.
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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:
Tap to reveal answer
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:
Tap to reveal answer
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:
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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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