Midpoint Formula - GMAT Quantitative Reasoning
Card 0 of 9
Consider segment 
with midpoint 
at the point 
.
I) Point 
has coordinates of 
.
II) Segment 
has a length of 
units.
What are the coordinates of point 
?
Consider segment
I) Point
II) Segment
What are the coordinates of point
In this case, we are given the midpoint of a line and asked to find one endpoint.
Statement I gives us the other endpoint. We can use this with midpoint formula (see below) to find our other point.
Midpoint formula: 
Statment II gives us the length of the line. However, we know nothing about its orientation or slope. Without some clue as to the steepness of the line, we cannot find the coordinates of its endpoints. You might think we can pull of something with distance formula, but there are going to be two unknowns and one equation, so we are out of luck.
So,
Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.
In this case, we are given the midpoint of a line and asked to find one endpoint.
Statement I gives us the other endpoint. We can use this with midpoint formula (see below) to find our other point.
Midpoint formula:
Statment II gives us the length of the line. However, we know nothing about its orientation or slope. Without some clue as to the steepness of the line, we cannot find the coordinates of its endpoints. You might think we can pull of something with distance formula, but there are going to be two unknowns and one equation, so we are out of luck.
So,
Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.
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Find endpoint 
given the following:
I) Segment 
has its midpoint at 
.
II) Point 
is located on the 
-axis, 
points from the origin.
Find endpoint
I) Segment
II) Point
Find endpoint Y given the following:
I) Segment RY has its midpoint at (45,65)
II) Point R is located on the x-axis, 13 points from the origin.
I) Gives us the location of the midpoint of our segment
(45,65)
II) Gives us the location of one endpoint
(13,0)
Use I) and II) to work backwards with midpoint formula to find the other endpoint.
So endpoint 
is at 
.
Therefore, both statements are needed to answer the question.
Find endpoint Y given the following:
I) Segment RY has its midpoint at (45,65)
II) Point R is located on the x-axis, 13 points from the origin.
I) Gives us the location of the midpoint of our segment
(45,65)
II) Gives us the location of one endpoint
(13,0)
Use I) and II) to work backwards with midpoint formula to find the other endpoint.
So endpoint
Therefore, both statements are needed to answer the question.
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Consider segment 
I) Endpoint 
is located at the origin
II) 
has a distance of 36 units
Where is endpoint 
located?
Consider segment
I) Endpoint
II)
Where is endpoint
To find the endpoint of a segment, we can generally use the midpoint formula; however, in this case we do not have enough information.
I) Gives us one endpoint
II) Gives us the length of DF
The problem is that we don't know the orientation of DF. It could go in infinitely many directions, so we can't find the location of 
without more information.
To find the endpoint of a segment, we can generally use the midpoint formula; however, in this case we do not have enough information.
I) Gives us one endpoint
II) Gives us the length of DF
The problem is that we don't know the orientation of DF. It could go in infinitely many directions, so we can't find the location of
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is the midpoint of line PQ. What are the coordinates of point P?
(1) Point Q is the origin.
(2) Line PQ is 8 units long.
(1) Point Q is the origin.
(2) Line PQ is 8 units long.
The midpoint formula is
,
with statement 1, we know that Q is 
and can solve for P:
and 
Statement 1 alone is sufficient.
Statement 2 doesn't provide enough information to solve for point P.
The midpoint formula is
with statement 1, we know that Q is
Statement 1 alone is sufficient.
Statement 2 doesn't provide enough information to solve for point P.
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In what quadrant or axis is the midpoint of the line segment with endpoints 
and 
located?
Statement 1: 
Statement 2: 
is in Quadrant IV.
In what quadrant or axis is the midpoint of the line segment with endpoints
Statement 1:
Statement 2:
The midpoint of the segment with endpoints 
and 
is 
.
If 
, then 
and 
, so the midpoint, having both of its coordinates positive, is in Quadrant I.
If 
is in Quadrant IV, then 
and 
. But the quadrant of the midpoint varies according to 
and 
:
Example 1: If 
, the midpoint is 
, or 
, putting it in Quadrant I.
Example 2: If 
, the midpoint is 
, or 
, putting it in Quadrant III.
Therefore, the first statement, but not the second, tells us all we need to know.
The midpoint of the segment with endpoints
If
If
Example 1: If
Example 2: If
Therefore, the first statement, but not the second, tells us all we need to know.
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A line segment has one of its endpoints at 
. In which quadrant, or on what axis, is its other endpoint?
Statement 1: The midpoint of the segment is 
.
Statement 2: The length of the segment is 10.
A line segment has one of its endpoints at
Statement 1: The midpoint of the segment is
Statement 2: The length of the segment is 10.
Statement 1 give us the means to find the other endpoint using the midpoint formula:
Similarly,
This makes the endpoint 
, which is in Quadrant I.
Statement 2 is also sufficient. 
, which is in Quadrant 1, is 12 units away from the nearest axis; since the length of the segment is 10, the entire segment must be in Quadrant I.
Statement 1 give us the means to find the other endpoint using the midpoint formula:
Similarly,
This makes the endpoint
Statement 2 is also sufficient.
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Consider segment 
. What are the coordinates of the midpoint of 
?
I) Point 
has coordinates of 
.
II) Point 
has coordinates of 
.
Consider segment
I) Point
II) Point
We are asked to find the midpoint of a line segment and given endpoints in our clues.
Midpoint formula is found by taking the average of the x and y values of two points.
We need both endpoints to solve this problem, so both statements are needed.
We are asked to find the midpoint of a line segment and given endpoints in our clues.
Midpoint formula is found by taking the average of the x and y values of two points.
We need both endpoints to solve this problem, so both statements are needed.
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Find the midpoint of segment 
given that point 
is at 
.
I) The 
coordinate of 
is twice that of 
, and the 
coordinate of 
is 
that of 
.
II) 
is 
units long.
Find the midpoint of segment
I) The
II)
To find the midpoint, we need to know both endpoints.
I) Gives us the means to find out other endpoint.
II) Gives us the length of PS, but we are not given any hint as to its orientation.Thus, we cannot find the other endpoint and we cannot find the midpoint.
Thus, Statement I alone is sufficient to answer the question.
To find the midpoint, we need to know both endpoints.
I) Gives us the means to find out other endpoint.
II) Gives us the length of PS, but we are not given any hint as to its orientation.Thus, we cannot find the other endpoint and we cannot find the midpoint.
Thus, Statement I alone is sufficient to answer the question.
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Find the midpoint of segment 
.
I) Endpoint 
has coordinates of 
.
II) Endpoint 
coordinate is half of 
, and 
coordinate is one sixteenth of 
coordinate.
Find the midpoint of segment
I) Endpoint
II) Endpoint
To find the midpoint of a segment we need both endpoints
I) Gives us one endpoint.
II) Gives us clues to find the other endpoint.
has coordinates of 
Use midpoint formula
To find the midpoint of a segment we need both endpoints
I) Gives us one endpoint.
II) Gives us clues to find the other endpoint.
Use midpoint formula
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