Midpoint Formula - GMAT Quantitative

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.

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.

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 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.

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.

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.

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 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

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.

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.

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.

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 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

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.

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.

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.

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.

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.

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.