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PSAT Math : Midpoint Formula

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Example Question #1 : How To Find The Endpoints Of A Line Segment

One endpoint of a line segment on the coordinate plane is (3,7) . The segment has length 10; the other endpoint has -coordinate 10 and is in Quadrant I. Give the -coordinate of the other endpoint. (Nearest tenth if applicable).

Possible Answers:

14.1

-0.1

-9.2

12.2

4.0

Correct answer:

14.1

Explanation:

Let be the -coordinate of the other endpoint.

The segment has endpoints (3,7) and (10, Y) .

Apply the distance formula

$d = \sqrt{(x_{2}-x_{1})^{2}+ (y_{2}-y_{1})^{2} }$

setting $d = 10, x_{1}= 3, x_{2} = 10, y_{1} = 7, y_{2}= Y$ ,

$10 = \sqrt{(10-3)^{2}+ (Y-7)^{2} }$

$100 =7^{2}+ (Y-7)^{2}$

$49+ (Y-7)^{2} = 100$

$(Y-7)^{2} = 51$

$Y-7 = \pm \sqrt{51}$

$Y = 7 \pm \sqrt{51}$

Therefore, there are two values of which fit the distance criterion.

One is $Y = 7 - \sqrt{51} \approx 7 - 7.1 \approx -0.1$ .

However, since the endpoint is in Quadrant I, it must have a positive coordinate, so this is eliminated as a choice.

The other is $Y = 7 + \sqrt{51} \approx 7 + 7.1 \approx 14.1$ , which is correct since it is positive.

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Example Question #2 : How To Find The Endpoints Of A Line Segment

The midpoint of a line segment is the point (1.4, -4.3) . One endpoint is (5.1, 3.1) ; give the -coordinate of the other.

Possible Answers:

1.85

-0.6

-2.3

-11.7

3.25

Correct answer:

-2.3

Explanation:

Let the endpoints of a line segment be

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

Then the midpoint of the segment will be

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

The -coordinate of the midpoint is $x_{m}= 1.4$ , and the -coordinate of the known endpoint is $x_{1} = 5.1$ , so we can solve for $x_{2}$ , the -coordinate of the unknown endpoint, in the following equation:

$\frac{x_{1}+x_{2} }{2} = x_{m}$

$\frac{5.1+x_{2} }{2} =1.4$

$5.1+x_{2} = 2.8$

$x_{2} = -2.3$

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Example Question #1 : How To Find The Endpoints Of A Line Segment

The midpoint of a line segment is the point (1.4, -4.3) . One endpoint is (5.1, 3.1) ; give the -coordinate of the other.

Possible Answers:

3.25

-0.6

-11.7

3.7

-2.3

Correct answer:

-11.7

Explanation:

Let the endpoints of a line segment be

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

Then the midpoint of the segment will be

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

The -coordinate of the midpoint is $y_{m}= -4.3$ , and the -coordinate of the known endpoint is $y_{1} = 3.1$ , so we can solve for $y_{2}$ , the -coordinate of the unknown endpoint, in the following equation:

$\frac{y_{1}+y_{2} }{2} = y_{m}$

$\frac{3.1+y_{2} }{2} =-4.3$

$3.1+x_{2} = -8.6$

$x_{2} = -11.7$

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Example Question #4 : How To Find The Endpoints Of A Line Segment

The midpoint of a line segment is the point $\left (4 \frac{3}{5},- 3 \frac{1 }{3} \right )$ . One endpoint is $\left (1 \frac{2}{3}, 1 \frac{2 }{5} \right )$ ; give the -coordinate of the other.

Possible Answers:

$3 \frac{2}{15}$

$7 \frac{8}{15}$

$5\frac{4}{15}$

$- 8\frac{1}{15}$

$10\frac{13}{15}$

Correct answer:

$- 8\frac{1}{15}$

Explanation:

Let the endpoints of a line segment be

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

Then the midpoint of the segment will be

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

The -coordinate of the midpoint is $y_{m}=- 3 \frac{1 }{3} = - \frac{10 }{3}$ , and the -coordinate of the known endpoint is $y_{1} = 1 \frac{2 }{5} = \frac{7}{5}$ ,

so we can solve for $y_{2}$ , the -coordinate of the unknown endpoint, in the following equation:

$\frac{ \frac{7}{5}+y_{2} }{2} = - \frac{10 }{3}$

$\frac{7}{5}+y_{2} = - \frac{20}{3}$

$y_{2} = - \frac{20}{3} - \frac{7}{5}$

$y_{2} = - \frac{100}{15} - \frac{21}{15}$

$y_{2} = - \frac{121}{15}$

$y_{2} = - 8\frac{1}{15}$

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Example Question #5 : How To Find The Endpoints Of A Line Segment

The midpoint of a line segment is the point $\left (4 \frac{3}{5},- 3 \frac{1 }{3} \right )$ . One endpoint is $\left (1 \frac{2}{3}, 1 \frac{2 }{5} \right )$ ; give the -coordinate of the other.

Possible Answers:

$7 \frac{8}{15}$

$5\frac{4}{15}$

$- 8\frac{1}{15}$

$3 \frac{2}{15}$

$10\frac{13}{15}$

Correct answer:

$7 \frac{8}{15}$

Explanation:

Let the endpoints of a line segment be

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

Then the midpoint of the segment will be

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

The -coordinate of the midpoint is $x_{m}= 4 \frac{3}{5} = \frac{23}{5}$ , and the -coordinate of the known endpoint is $x_{1} = 1 \frac{2}{3} = \frac{5}{3}$ , so we can solve for $x_{2}$ , the -coordinate of the unknown endpoint, in the following equation:

$\frac{x_{1}+x_{2} }{2} = x_{m}$

$\frac{\frac{5}{3}+x_{2} }{2} = \frac{23}{5}$

$\frac{5}{3}+x_{2} = \frac{46}{5}$

$x_{2} = \frac{46}{5} - \frac{5}{3}$

$x_{2} = \frac{138}{15} - \frac{25}{15}$

$x_{2} = \frac{113}{15}$

$x_{2} = 7 \frac{8}{15}$

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PSAT Math : Midpoint Formula

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : How To Find The Endpoints Of A Line Segment

Example Question #2 : How To Find The Endpoints Of A Line Segment

Example Question #1 : How To Find The Endpoints Of A Line Segment

Example Question #4 : How To Find The Endpoints Of A Line Segment

Example Question #5 : How To Find The Endpoints Of A Line Segment

All PSAT Math Resources

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