Lines - GMAT Quantitative
Card 0 of 603
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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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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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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A line segment has an endpoint at 
; what is its length?
-
Its other endpoint is 
-
Its midpoint is 
A line segment has an endpoint at
-
Its other endpoint is
-
Its midpoint is
Given the other endpoint, you can use the distance formula to find the length of the segment:
Given the midpoint, you can use the distance formula to find the distance from the first endpoint to the midpoint, then double that to get the length of the segment:
The total length is twice that, or 10.
The answer is that either statement alone is sufficient to answer the question.
Given the other endpoint, you can use the distance formula to find the length of the segment:
Given the midpoint, you can use the distance formula to find the distance from the first endpoint to the midpoint, then double that to get the length of the segment:
The total length is twice that, or 10.
The answer is that either statement alone is sufficient to answer the question.
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Note: Figure NOT drawn to scale.
Give 
.
Statement 1: 
Statement 2: 
Note: Figure NOT drawn to scale.
Give
Statement 1:
Statement 2:
If you know only that 
, then you know that 
and 
, but you still need 
, or a way finding it.
If you know only that 
, you still know only that 
, but you don't know their actual lengths.
If you know both facts, then you know 
If you know only that
If you know only that
If you know both facts, then you know
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Consider segment 
.
I) Point 
can be found at the point 
.
II) Segment 
had a length of 
units.
Find the coordinates of point 
.
Consider segment
I) Point
II) Segment
Find the coordinates of point
Statement I gives us a point.
Statement II gives us the length of the segment.
We are asked to find the coordinates of the other end of the segment. However, we will need more information. Even with all of our information, we have no clue as to the orientation of the line. It could be 14 units straight up and down, it could be a perfectly horizontal line, or something inbetween, thus our answer is:
Neither I nor II are sufficient to answer the question. More information is needed.
Statement I gives us a point.
Statement II gives us the length of the segment.
We are asked to find the coordinates of the other end of the segment. However, we will need more information. Even with all of our information, we have no clue as to the orientation of the line. It could be 14 units straight up and down, it could be a perfectly horizontal line, or something inbetween, thus our answer is:
Neither I nor II are sufficient to answer the question. More information is needed.
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Find the length of Segment YZ
I) Point Y is located at the point 
.
II) Point Z has a y-coordinate twice that of Point Y and an x-coordinate one-third of Point Y.
Find the length of Segment YZ
I) Point Y is located at the point
II) Point Z has a y-coordinate twice that of Point Y and an x-coordinate one-third of Point Y.
To find the length of a segment, use the distance formula. The distance formula is given by the following:
Where your 
's and 
's correspond to the coordinates of the endpoints.
To find the length of Segment YZ, we need the endpoints.
Statement I gives you Point Y's coordinates.
Statement II relates Point Z's coordinates to Point Y's coordinates. Thus, we can find the point Z using Statement II.
Therefore, we need both.
Recap:
Find the length of Segment YZ
I) Point Y is located at the point 
.
II) Point Z has a y-coordinate twice that of Point Y, and an x-coordinate one-third of Point Y.
Use Statement II along with Statement I to find the coordinates of Point Z:
Then, use distance formula to find the length of Segment YZ:
To find the length of a segment, use the distance formula. The distance formula is given by the following:
Where your
To find the length of Segment YZ, we need the endpoints.
Statement I gives you Point Y's coordinates.
Statement II relates Point Z's coordinates to Point Y's coordinates. Thus, we can find the point Z using Statement II.
Therefore, we need both.
Recap:
Find the length of Segment YZ
I) Point Y is located at the point
II) Point Z has a y-coordinate twice that of Point Y, and an x-coordinate one-third of Point Y.
Use Statement II along with Statement I to find the coordinates of Point Z:
Then, use distance formula to find the length of Segment YZ:
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Consider segment 
I) Endpoint 
is located at the point 
.
Ii) Endpoint 
has an x-coordinate twice that of 
and a y-coordinate 15 times that of H.
What is the length of 
?
Consider segment
I) Endpoint
Ii) Endpoint
What is the length of
To find the length of a segment, we need both endpoints.
Statement I gives us one endpoint.
Statement II relates 
and 
, allowing us to find the second endpoint.
Thus, we need both. Once we have both endpoints, distance is easily calculated via the distance formula or the Pythagorean theorem.
Using Statement II, we find the second endpoint to be 
. Use the distance formula to find your answer:
To find the length of a segment, we need both endpoints.
Statement I gives us one endpoint.
Statement II relates
Thus, we need both. Once we have both endpoints, distance is easily calculated via the distance formula or the Pythagorean theorem.
Using Statement II, we find the second endpoint to be
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Find the equation to a line perpendicular to line 
.
- The slope of line
is 
.
- Line
goes through point 
.
Find the equation to a line perpendicular to line
- The slope of line
- Line
Statement 1: Since the line we're looking for is perpendicular to line XY, our slope will be the inverse of line XY's slope 
.
The slope of our line is then 
. Just knowing the slope however, is not sufficient information to answer the question.
Statement 2: We're provided with a point which will allow us the write the equation.
Statement 1: Since the line we're looking for is perpendicular to line XY, our slope will be the inverse of line XY's slope
The slope of our line is then
Statement 2: We're provided with a point which will allow us the write the equation.
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Calculate the equation of a line perpendicular to line 
.
- The equation for line
is 
.
- Line
goes through point 
.
Calculate the equation of a line perpendicular to line
- The equation for line
- Line
Statement 1: We're given the equation to line AB which contains the slope. Because the line we're being asked for is perpendicular to it, we know the slope will be its inverse.
The slope of our line is then 
Statement 2: We can write the equation to the perpendicular line only if we have a point that falls within that line. Luckily, we're given such a point in statement 2.
Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.
Statement 1: We're given the equation to line AB which contains the slope. Because the line we're being asked for is perpendicular to it, we know the slope will be its inverse.
The slope of our line is then
Statement 2: We can write the equation to the perpendicular line only if we have a point that falls within that line. Luckily, we're given such a point in statement 2.
Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.
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Find the equation of the line perpendicular to 
.
I) 
has a slope of 
.
II) The line must pass through the point 
.
Find the equation of the line perpendicular to
I)
II) The line must pass through the point
Find the equation of the line perpendicular to r(x)
I) r(x) has a slope of -15
II) The line must pass through the point (9, 96)
Recall that perpendicular lines have opposite reciprocal slopes.
Use I) to find the slope of our new line
Use II) along with our slope to find the y-intercept of our new line.
Therefore both statements are needed.
Find the equation of the line perpendicular to r(x)
I) r(x) has a slope of -15
II) The line must pass through the point (9, 96)
Recall that perpendicular lines have opposite reciprocal slopes.
Use I) to find the slope of our new line
Use II) along with our slope to find the y-intercept of our new line.
Therefore both statements are needed.
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Consider 
:
Find 
, a line perpendicular to 
, given the following:
I) 
passes through the point 
.
II) 
passes through the point 
.
Consider
Find
I)
II)
Recall that perpendicular lines have opposite, reciprocal slopes. We can find the slope of 
from the question.
Statement I gives us a point on 
, which we can use to find the y-intercept of 
, and then the equation.
The slope of 
must be the opposite reciprocal of 
, this makes our slope 
.
Statement I tells us that 
passes through the point 
, so we can use slope-intercept form to find our equation:
So, our equation is
Statement II gives us a point on 
, which does not help us in the slightest with 
. Therefore, only Statement I is sufficient.
Recall that perpendicular lines have opposite, reciprocal slopes. We can find the slope of
Statement I gives us a point on
The slope of
Statement I tells us that
So, our equation is
Statement II gives us a point on
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Give the equation of a line on the coordinate plane.
Statement 1: The line shares an 
-intercept and its 
-intercept with the line of the equation 
.
Statement 2: The line is perpendicular to the line of the equation 
.
Give the equation of a line on the coordinate plane.
Statement 1: The line shares an
Statement 2: The line is perpendicular to the line of the equation
Assume Statement 1 alone. The 
-intercept of the line of the equation can be found by substituting 
and solving for 
:
The 
-intercept of the line is the origin 
; it follows that this is also the 
-intercept.
Therefore, Statement 1 alone yields only one point of the line, from which its equation cannot be determined.
Assume Statement 2 alone. The slope of the line of the equation 
can be calculated by putting it in slope-intercept form 
:
The slope of this line is the coefficient of 
, which is 
. A line perpendicular to this one has as its slope the opposite of the reciprocal of 
, which is
.
However, there are infintely many lines with this slope, so no further information can be determined.
Now assume both statements to be true. From Statement 1, the slope of the line is 
, and from Statement 2, the 
-coordinate of the 
-initercept is 
. Substitute in the slope-intercept form:
Assume Statement 1 alone. The
The
Therefore, Statement 1 alone yields only one point of the line, from which its equation cannot be determined.
Assume Statement 2 alone. The slope of the line of the equation
The slope of this line is the coefficient of
However, there are infintely many lines with this slope, so no further information can be determined.
Now assume both statements to be true. From Statement 1, the slope of the line is
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Consider 
and 
.
Find the slope of 
.
I) 
passes through the point 
.
II) 
is perpendicular to 
.
Consider
Find the slope of
I)
II)
We are given a line, f(x), and asked to find the slope of another line, h(x).
I) Gives a point on h(x). We could plug in the point and solve for our slope. When we do this since x=0 we are unable to find the value for our slope. Therefore, statement I is not sufficient to solve the question.
II) Tells us the two lines are perpendicular. Take the opposite reciprocal of the slope of f(x) to find the slope of h(x).
Therefore,
and thus the slope of h(x) will be,
.
Statement II is sufficient to answer the question.
We are given a line, f(x), and asked to find the slope of another line, h(x).
I) Gives a point on h(x). We could plug in the point and solve for our slope. When we do this since x=0 we are unable to find the value for our slope. Therefore, statement I is not sufficient to solve the question.
II) Tells us the two lines are perpendicular. Take the opposite reciprocal of the slope of f(x) to find the slope of h(x).
Therefore,
and thus the slope of h(x) will be,
Statement II is sufficient to answer the question.
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Calculate the slope of a line perpendicular to line 
.
- Line
passes through points 
and 
.
- The equation for line
is 
.
Calculate the slope of a line perpendicular to line
- Line
- The equation for line
Statement 1: We can use the points provided to find the slope of line AB.
Since the slope we're being asked for is of a line perpendicular to line AB, their slopes are inverses of each other.
The slope of our line is then 
Statement 2: Since we're provided with the line's equation, we just need to look for the slope.
Where 
is the slope and 
is the y-intercept.
In this case, we have 
so 
. Because our line is perpendicular to line AB, the slope we're looking for is 
Statement 1: We can use the points provided to find the slope of line AB.
Since the slope we're being asked for is of a line perpendicular to line AB, their slopes are inverses of each other.
The slope of our line is then
Statement 2: Since we're provided with the line's equation, we just need to look for the slope.
Where
In this case, we have
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