GMAT Math : Coordinate Geometry

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Example Question #1 : Coordinate Geometry

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

Possible Answers:

Quadrants III and IV

Quadrants I and III

Quadrants II and III

Quadrants II and IV

Quadrants I and IV

Correct answer:

Quadrants II and IV

Explanation:

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

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

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Example Question #2 : Coordinate Geometry

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

Possible Answers:

$\displaystyle 2$

$\displaystyle 8$

It cannot be determined from the information given.

$\displaystyle 5$

$\displaystyle 10$

Correct answer:

It cannot be determined from the information given.

Explanation:

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

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

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

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

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

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

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

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

$\displaystyle A + B + 6 = 16$

$\displaystyle A + B = 10$

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

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

$\displaystyle A + B = 10$

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) $\displaystyle (2, 9)$ and (4, 3) $\displaystyle (4, 3)$.

Possible Answers:

(6,12) $\displaystyle (6,12)$

$\displaystyle (2.5,5.5)$

(2,5) $\displaystyle (2,5)$

(-3,6) $\displaystyle (-3,6)$

(3,6) $\displaystyle (3,6)$

Correct answer:

(3,6) $\displaystyle (3,6)$

Explanation:

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

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

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Example Question #2 : Coordinate Geometry

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

Possible Answers:

(-3,6) $\displaystyle (-3,6)$

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

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

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

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

Correct answer:

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

Explanation:

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

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

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Example Question #4 : Coordinate Geometry

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

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

Possible Answers:

$\small (74,40)$ $\displaystyle \small (74,40)$

$\small \small (74,35)$ $\displaystyle \small \small (74,35)$

$\small \small (75,45)$ $\displaystyle \small \small (75,45)$

$\small \small (73,42)$ $\displaystyle \small \small (73,42)$

$\small \small (70,35)$ $\displaystyle \small \small (70,35)$

Correct answer:

$\small (74,40)$ $\displaystyle \small (74,40)$

Explanation:

Midpoint formula is as follows:

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

Plug in and calculate:

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

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Example Question #4 : Coordinate Geometry

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

Possible Answers:

(-1,17) $\displaystyle (-1,17)$

(17,-1) $\displaystyle (17,-1)$

(5,17) $\displaystyle (5,17)$

$\displaystyle (-5,-17)$

(17,5) $\displaystyle (17,5)$

Correct answer:

(-1,17) $\displaystyle (-1,17)$

Explanation:

Midpoints can be found using the following:

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

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

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

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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 $\displaystyle AB$ if A=(2,3) $\displaystyle A=(2,3)$ and $\displaystyle B=(2,-7)?$

Possible Answers:

(2,4) $\displaystyle (2,4)$

(0,4) $\displaystyle (0,4)$

(0,-10) $\displaystyle (0,-10)$

(2,-2) $\displaystyle (2,-2)$

(0,-2) $\displaystyle (0,-2)$

Correct answer:

(2,-2) $\displaystyle (2,-2)$

Explanation:

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

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Example Question #3 : Lines

The quadrilateral with vertices $\displaystyle (0,0),(22,0), (16,4), (4,4)$ is a trapezoid. What are the endpoints of its midsegment?

Possible Answers:

$\displaystyle (11,0),(10,4)$

$\displaystyle (2,2),(19,2 )$

$\displaystyle (0,0), (16,4)$

$\displaystyle (22,0), (4,4)$

$\displaystyle (0,2),(23,2 )$

Correct answer:

$\displaystyle (2,2),(19,2 )$

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 $\displaystyle (0,0), (4,4)$ and $\displaystyle (22,0), (16,4)$. The midpoint of each can be found by taking the means of the $\displaystyle x$- and $\displaystyle y$-coordinates:

$(0,0), (4,4): \left ( \frac{0+4}{2},\frac{0+4}{2} \right ) \Rightarrow (2,2)$ $\displaystyle (0,0), (4,4): \left ( \frac{0+4}{2},\frac{0+4}{2} \right ) \Rightarrow (2,2)$

$(22,0), (16,4): \left ( \frac{22+16}{2},\frac{0+4}{2} \right ) \Rightarrow (19,2)$ $\displaystyle (22,0), (16,4): \left ( \frac{22+16}{2},\frac{0+4}{2} \right ) \Rightarrow (19,2)$

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

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Example Question #3 : Coordinate Geometry

The midpoint of a line segment with endpoints (2A, B) $\displaystyle (2A, B)$ and $\displaystyle (B + 7, 3A - 2)$ is (9, 6) $\displaystyle (9, 6)$. What is $\displaystyle A$?

Possible Answers:

$\displaystyle 7$

It cannot be determined from the information given.

$\displaystyle 5$

$\displaystyle 6$

$\displaystyle 3$

Correct answer:

$\displaystyle 3$

Explanation:

If the midpoint of a line segment with endpoints (2A, B) $\displaystyle (2A, B)$ and $\displaystyle (B + 7, 3A - 2)$ is (9, 6) $\displaystyle (9, 6)$, then by the midpoint formula,

$\frac{2A + (B + 7)}{2} = 9$ $\displaystyle \frac{2A + (B + 7)}{2} = 9$

and

$\frac{B + (3A - 2)}{2} = 6$ $\displaystyle \frac{B + (3A - 2)}{2} = 6$.

The first equation can be simplified as follows:

$\displaystyle 2A + B + 7 = 18$

$\displaystyle 2A + B = 11$

The second can be simplified as follows:

$\displaystyle 3A + B - 2 = 12$

$\displaystyle 3A + B = 14$

This is a system of linear equations. $\displaystyle A$ can be calculated by subtracting:

$\displaystyle 3A + B = 14$

$\underline{2A + B = 11 }$ $\displaystyle \underline{2A + B = 11 }$

$A \; \; \; \; \; \; \; \; = 3$ $\displaystyle A \; \; \; \; \; \; \; \; = 3$

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Example Question #5 : Lines

If the midpoint of $\overline{VU}$ $\displaystyle \overline{VU}$ is $\left ( 6,7\right )$ $\displaystyle \left ( 6,7\right )$ and $\textup{Point }V$ $\displaystyle \textup{Point }V$ is at $\left (2,9\right )$ $\displaystyle \left (2,9\right )$, what are the coordinates of $\textup{Point }U$ $\displaystyle \textup{Point }U$?

Possible Answers:

$\left ( 10,5\right )$ $\displaystyle \left ( 10,5\right )$

$\left ( -5,10\right )$ $\displaystyle \left ( -5,10\right )$

$\left ( 5,10\right )$ $\displaystyle \left ( 5,10\right )$

$\left ( 14,23\right )$ $\displaystyle \left ( 14,23\right )$

$\left ( 23,14\right )$ $\displaystyle \left ( 23,14\right )$

Correct answer:

$\left ( 10,5\right )$ $\displaystyle \left ( 10,5\right )$

Explanation:

Midpoint formula is as follows:

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

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)

$\small 6=\frac{2+x'}{2}$ $\displaystyle \small 6=\frac{2+x'}{2}$ and $\small 7=\frac{9+y'}{2}$ $\displaystyle \small 7=\frac{9+y'}{2}$

$\small x'=6*2-2=10$ $\displaystyle \small x'=6*2-2=10$ $\small y'=7*2-9=14-9=5$ $\displaystyle \small y'=7*2-9=14-9=5$

So we have (10,5) as point U

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GMAT Math : Coordinate Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Coordinate Geometry

Example Question #2 : 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

Example Question #3 : Lines

Example Question #3 : Coordinate Geometry

Example Question #5 : Lines

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