# GMAT Math : Calculating the midpoint of a line segment

## Example Questions

### Example Question #1 : Coordinate Geometry

Which of the following quadrants can contain the midpoint of a line segment with endpoints  and  for some nonzero value of ?

Explanation:

The midpoint of the line segment with endpoints  and   is , or

If , then the -coordinate is negative and the -coordinate is positive, so the midpoint is in Quadrant II. If , the reverse is true, so the midpoint is in Quadrant IV.

### Example Question #1 : Coordinate Geometry

The midpoint of a line segment with endpoints  and  is . Sove for .

It cannot be determined from the information given.

It cannot be determined from the information given.

Explanation:

The midpoint of a line segment with endpoints  is

.

Substitute the coordinates of the endpoints, then set each equation to the appropriate midpoint coordinate.

-coordinate:

-coordinate:

Simplify each, then solve the system of linear equations in two variables:

The two linear equations turn out to be equivalent, meaning that there are infinitely many solutions to the system. Therefore, insufficient information is given to answer the question.

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

Find the midpoint of the points  and .

Explanation:

Add the corresponding points together and divide both values by 2:

### Example Question #1 : Coordinate Geometry

What is the midpoint of and ?

Explanation:

Add the x-values and divide by 2, and then add the y-values and divide by 2.  Be careful of the negatives!

### Example Question #5 : Coordinate Geometry

Consider segment  which passes through the points  and .

What are the correct coordinates for the midpoint of ?

Explanation:

Midpoint formula is as follows:

Plug in and calculate:

### Example Question #6 : Coordinate Geometry

Segment  has endpoints of  and . If the midpoint of  is given by point , what are the coordinates of point ?

Explanation:

Midpoints can be found using the following:

Plug in our points (-6,8) and (4,26) to find the midpoint.

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

What are the coordinates of the mipdpoint of the line segment  if  and