GMAT Math : Calculating the endpoints of a line segment

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Example Question #8 : Coordinate Geometry

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

Possible Answers:

(0,2),(23,2 )

(11,0),(10,4)

(2,2),(19,2 )

(0,0), (16,4)

(22,0), (4,4)

Correct answer:

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

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

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

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

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Example Question #1 : Calculating The Endpoints Of A Line Segment

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

Possible Answers:

It cannot be determined from the information given.

Correct answer:

Explanation:

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

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

and

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

The first equation can be simplified as follows:

2A + B + 7 = 18

2A + B = 11

The second can be simplified as follows:

3A + B - 2 = 12

3A + B = 14

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

3A + B = 14

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

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

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Example Question #1 : Coordinate Geometry

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

Possible Answers:

$\left ( 14,23\right )$

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

$\left ( 10,5\right )$

$\left ( 23,14\right )$

$\left ( 5,10\right )$

Correct answer:

$\left ( 10,5\right )$

Explanation:

Midpoint formula is as follows:

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

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)

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

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

So we have (10,5) as point U

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Example Question #1 : Calculating The Endpoints Of A Line Segment

A line segment has its midpoint at (-1,3) and an endpoint at (3,8) . What are the coordinates of the other endpoint?

Possible Answers:

(-3,-1)

(-3,-8)

(-2,1)

(-5,-2)

(-2,-5)

Correct answer:

(-5,-2)

Explanation:

Because we are given the midpoint and one of the endpoints, we know the x coordinate of the other endpoint will be the same distance away from the midpoint in the x direction, and the y coordinate of the other endpoint will be the same distance away from the midpoint in the y direction. Given two endpoints of the form:

(x_1,y_1),(x_2,y_2)

The midpoint of these two endpoints has the coordinates:

$(x_m,y_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Plugging in values for the given midpoint and one of the endpoints, which we can see is (x_2,y_2) because it lies to the right of the midpoint, we can solve for the other endpoint as follows:

$(x_m,y_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$(-1,3)=(\frac{x_1+3}{2},\frac{y_1+8}{2})$

$\frac{x_1+3}{2}=-1\rightarrow x_1+3=-2\rightarrow x_1=-5$

$\frac{y_1+8}{2}=3\rightarrow y_1+8=6\rightarrow y_1=-2$

So the other endpoint has the coordinates (-5,-2)

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Example Question #2 : Calculating The Endpoints Of A Line Segment

Consider segment with endpoint at (8,9) . If the midpoint of can be found at (-15, -67) , what are the coordinates of point ?

Possible Answers:

(-46,-150)

(-38,-143)

(143,38)

(-143,-38)

(38,143)

Correct answer:

(-38,-143)

Explanation:

Recall midpoint formula:

$(x',y')=\left(\frac{x^1+x^2}{2}, \frac{y^1+y^2}{2}\right)$

In this case we have (x'y') and one of our other (x,y) points.

Plug and chug:

$(-15,-67)=\left(\frac{8+x}{2}, \frac{9+y}{2}\right)$

If you make this into two equations and solve you get the following.

$\small -15=\frac{8+x}{2}, x=-38$

$\small -67=\frac{9+y}{2}, y=-143$

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GMAT Math : Calculating the endpoints of a line segment

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #8 : Coordinate Geometry

Example Question #1 : Calculating The Endpoints Of A Line Segment

Example Question #1 : Coordinate Geometry

Example Question #1 : Calculating The Endpoints Of A Line Segment

Example Question #2 : Calculating The Endpoints Of A Line Segment

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