### All GMAT Math Resources

## 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 ?

**Possible Answers:**

Quadrants II and IV

Quadrants I and IV

Quadrants I and III

Quadrants III and IV

Quadrants II and III

**Correct answer:**

Quadrants II and IV

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 : Coordinate Geometry

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

**Possible Answers:**

It cannot be determined from the information given.

**Correct answer:**

It cannot be determined from the information given.

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 : Coordinate Geometry

Find the midpoint of the points and .

**Possible Answers:**

**Correct answer:**

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

### Example Question #4 : Coordinate Geometry

What is the midpoint of and ?

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

Midpoints can be found using the following:

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

### Example Question #7 : Coordinate Geometry

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

**Possible Answers:**

**Correct answer:**

The midpoint formula is

### Example Question #8 : Coordinate Geometry

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

**Possible Answers:**

**Correct answer:**

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 : Coordinate Geometry

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

**Possible Answers:**

It cannot be determined from the information given.

**Correct answer:**

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 : Coordinate Geometry

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

**Possible Answers:**

**Correct answer:**

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