# GMAT Math : Coordinate Geometry

## Example Questions

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### Example Question #1 : Midpoint Formula

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 #2 : Midpoint Formula

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 #3 : Midpoint Formula

Find the midpoint of the points  and .

Explanation:

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

### Example Question #4 : Midpoint Formula

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 : Midpoint Formula

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 : Midpoint Formula

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 #7 : Midpoint Formula

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

Explanation:

The midpoint formula is

### Example Question #8 : Midpoint Formula

The quadrilateral with vertices  is a trapezoid. What are the endpoints of its midsegment?

Explanation:

The midsegment of a trapezoid is the segment whose endpoints are the midpoints of its legs - its nonparallel opposite sides. These two sides are the ones with endpoints  and . The midpoint of each can be found by taking the means of the - and -coordinates:

The midsegment is the segment that has endpoints (2,2) and (19,2)

### Example Question #9 : Midpoint Formula

The midpoint of a line segment with endpoints  and  is . What is ?

It cannot be determined from the information given.

Explanation:

If the midpoint of a line segment with endpoints  and  is , then by the midpoint formula,

and

.

The first equation can be simplified as follows:

or

The second can be simplified as follows:

or

This is a system of linear equations.  can be calculated by subtracting:

### Example Question #10 : Midpoint Formula

If the midpoint of  is  and  is at , what are the coordinates of ?

Explanation:

Midpoint formula is as follows:

In this case, we have x,y and the value of the midpoint. We need to findx' and y'

V is at (2,9) and the midpoint is at (6,7)

and

So we have (10,5) as point U

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