How to find midpoint of a line segment in an XY plane?

How do you handle GRE quant questions that ask you about midpoint of a line segment in an XY plane?

The mid-point of two points A and B is a point on the line segment that joins the two points and that is equidistant from both of them. In the following figure, M is the midpoint of line segment AB such that M lies on the segment AB and MA=MB.

If  and  are two points on the X-Y plane, then the coordinate of the midpoint, say  can be computed as follows:

In other words the coordinates of the midpoint of a line segment is the arithmetic mean of the X and Y coordinates of the end points of the segment. 

If the X coordinates of both the points are same, the midpoint will have the same X coordinate. If the Ycoordinates of both the points are same, the midpoint will have the same Y coordinate.

The mid-point questions on the GRE are usually simple. Most of them expect you to know the mid-point formula. Let us look at an example: 

Example:

What is the value of  , if  is the midpoint of segment  and coordinates of point  is  and point  is  ?

A) -15

B) -9

C) -5

D) -3

E) 3

Solution:

Step 1: Here we know the coordinates of the midpoint and need to find out the X coordinate of point  . The information for Y coordinates is redundant i.e. not needed to solve this problem. The midpoint formula is simple:

The x coordinate of midpoint  = 

Step 2: We need to solve the above equation for 

Using cross-multiplication we get: 

Isolating variable we get: 

The correct answer is B

On the GRE, more complex questions may involve concepts such as slope of the line, parallel or perpendicular lines or the equation of a line.