GMAT Math : Calculating the midpoint of a line segment

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Example Question #831 : Problem Solving Questions

Which of the following quadrants can contain the midpoint of a line segment with endpoints (0,A) and (-A,0) for some nonzero value of ?

Possible Answers:

Quadrants I and IV

Quadrants II and IV

Quadrants I and III

Quadrants III and IV

Quadrants II and III

Correct answer:

Quadrants II and IV

Explanation:

The midpoint of the line segment with endpoints (0,A) and (-A,0) is $\left ( \frac{0+ (-A) }{2}, \frac{A+ 0 }{2}, \right )$ , or $\left (- \frac{A }{2}, \frac{A }{2}, \right )$

If A > 0 , then the -coordinate is negative and the -coordinate is positive, so the midpoint is in Quadrant II. If A < 0 , the reverse is true, so the midpoint is in Quadrant IV.

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Example Question #1 : Coordinate Geometry

The midpoint of a line segment with endpoints (A + 2, 2B-1) and (B+4, 2A+1) is (8,10) . Sove for .

Possible Answers:

It cannot be determined from the information given.

Correct answer:

It cannot be determined from the information given.

Explanation:

The midpoint of a line segment with endpoints $(x_{1}, y_{1}), (x_{2}, y_{2})$ is

$\left ( \frac{x_{1}+x_{2} }{2},\frac{y_{1}+y_{2} }{2} \right )$ .

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

-coordinate: $\frac{(A + 2)+ (B + 4) }{2} = 8$

-coordinate: $\frac{(2B-1)+(2A +1) }{2} = 10$

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

$\frac{(A + 2)+ (B + 4) }{2} = 8$

$\frac{(A + B + 6) }{2} = 8$

A + B + 6 = 16

A + B = 10

$\frac{(2B-1)+(2A +1) }{2} = 10$

$\frac{2A + 2B }{2} = 10$

A + B = 10

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.

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Example Question #1 : Coordinate Geometry

Find the midpoint of the points (2, 9) and (4, 3) .

Possible Answers:

(2,5)

(2.5,5.5)

(-3,6)

(3,6)

(6,12)

Correct answer:

(3,6)

Explanation:

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

$(\frac{2+4}{2},\frac{9+3}{2}) = (3, 6)$

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Example Question #2 : Coordinate Geometry

What is the midpoint of (-5,1) and (1,4) ?

Possible Answers:

$(3,\frac{5}{2})$

$(3,\frac{3}{2})$

(-3,6)

$(-2,\frac{5}{2})$

$(2,\frac{5}{2})$

Correct answer:

$(-2,\frac{5}{2})$

Explanation:

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

$(\frac{-5+1}{2},\frac{1+4}{2}) = (-2, \frac{5}{2})$

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Example Question #4 : Coordinate Geometry

Consider segment $\overline{JK}$ which passes through the points $\left ( 4,5\right )$ and $\left ( 144,75\right )$ .

What are the correct coordinates for the midpoint of $\overline{JK}$ ?

Possible Answers:

$\small (74,40)$

$\small \small (74,35)$

$\small \small (75,45)$

$\small \small (73,42)$

$\small \small (70,35)$

Correct answer:

$\small (74,40)$

Explanation:

Midpoint formula is as follows:

$\small m=(\frac{x+x'}{2},\frac{y+y'}{2})$

Plug in and calculate:

$\small \small \small m=(\frac{4+144}{2},\frac{75+5}{2})=(74,40)$

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Example Question #4 : Coordinate Geometry

Segment has endpoints of (-6,8) and (4,26) . If the midpoint of is given by point , what are the coordinates of point ?

Possible Answers:

(-1,17)

(17,-1)

(5,17)

(-5,-17)

(17,5)

Correct answer:

(-1,17)

Explanation:

Midpoints can be found using the following:

$midpoint(x,y)=\left(\frac{x+x'}{2},\frac{y+y'}{2}\right)$

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

$\small \small midpoint(x,y)=\left(\frac{-6+4}{2},\frac{8+26}{2}\right)=\left(\frac{-2}{2},\frac{34}{2}\right)=(-1,17)$

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Example Question #3 : Calculating The Midpoint Of A Line Segment

What are the coordinates of the mipdpoint of the line segment if A=(2,3) and B=(2,-7)?

Possible Answers:

(2,4)

(0,4)

(0,-10)

(2,-2)

(0,-2)

Correct answer:

(2,-2)

Explanation:

The midpoint formula is $(\frac{x_{2}+x_{1}}{2},\frac{y_{2}+y_{1}}{2}) = (\frac{2+2}{2},\frac{3+-7}{2})=(2,-2).$

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GMAT Math : Calculating the midpoint of a line segment

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #831 : Problem Solving Questions

Example Question #1 : Coordinate Geometry

Example Question #1 : Coordinate Geometry

Example Question #2 : Coordinate Geometry

Example Question #4 : Coordinate Geometry

Example Question #4 : Coordinate Geometry

Example Question #3 : Calculating The Midpoint Of A Line Segment

All GMAT Math Resources

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