GMAT Math : Midpoint Formula

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #151 : Coordinate Geometry

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 ?

Possible Answers:

Neither I nor II are sufficient to answer the question. More information is needed.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Either statement alone is sufficient to answer the question.

Both statements are necessary to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Correct answer:

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Explanation:

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.

Example Question #2 : Dsq: Calculating The Endpoints Of A Line Segment

Find endpoint  given the following:

I) Segment  has its midpoint at .

II) Point  is located on the -axis,  points from the origin.

Possible Answers:

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Either statement is sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

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.

Example Question #3 : Dsq: Calculating The Endpoints Of A Line Segment

Consider segment 

I) Endpoint  is located at the origin

II)  has a distance of 36 units

Where is endpoint  located?

Possible Answers:

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Correct answer:

Neither statement is sufficient to answer the question. More information is needed.

Explanation:

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.

Example Question #2 : Dsq: Calculating The Endpoints Of A Line Segment

 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.

Possible Answers:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. 

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient. 

Statements (1) and (2) TOGETHER are NOT sufficient.

Correct answer:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. 

Explanation:

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.

Example Question #1 : Dsq: Calculating The Midpoint Of A Line Segment

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.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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.

 

Example Question #2 : Dsq: Calculating The Midpoint Of A Line Segment

In what quadrant or axis is the midpoint of the line segment with endpoints  and  located?

Statement 1: 

Statement 2:  is in Quadrant IV.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

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.

Example Question #3 : Dsq: Calculating The Midpoint Of A Line Segment

Consider segment . What are the coordinates of the midpoint of ?

I) Point  has coordinates of .

II) Point  has coordinates of .

 

Possible Answers:

Statement 2 is sufficient to solve the question, but statement 1 is not sufficient to solve the question.

Neither statement is sufficient to solve the question. More information is needed.

Both statements taken together are sufficient to solve the question.

Statement 1 is sufficient to solve the question, but statement 2 is not sufficient to solve the question.

Each statement alone is enough to solve the question.

Correct answer:

Both statements taken together are sufficient to solve the question.

Explanation:

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. 

Example Question #71 : Lines

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.

 

Possible Answers:

Neither statement is sufficient to answer the question. More information is needed.

Both statements together are needed to answer the question.

Either statement alone is sufficient to answer the question.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Correct answer:

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Explanation:

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.

Example Question #5 : Dsq: Calculating The Midpoint Of A Line Segment

Find the midpoint of segment .

I) Endpoint  has coordinates of .

II) Endpoint   coordinate is half of , and   coordinate is one sixteenth of   coordinate.

Possible Answers:

Neither statement is sufficient to answer the question. More information is needed.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

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

 

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