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Intermediate Geometry : Coordinate Geometry

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Example Questions

Example Question #21 : Midpoint Formula

Find the midpoint of a line segment that has endpoints at $\left(\frac{1}{2}, 5\right)$ and $\left(-\frac{3}{2}, 1\right)$ .

Possible Answers:

$(-\frac{3}{4}, 6)$

$(-\frac{1}{2}, -\frac{1}{2})$

$(-\frac{1}{2}, 2)$

$(-\frac{1}{2}, 3)$

Correct answer:

$(-\frac{1}{2}, 3)$

Explanation:

Recall the formula for finding a midpoint of a line segment:

$\text{Midpoint}=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

The coordinates of the midpoint is just the average of the x-coordinates and the average of the y-coordinates.

Plug in the given points to find the midpoint of the line segment.

$\text{Midpoint}=(\frac{\frac{1}{2}-\frac{3}{2}}{2}, \frac{5+1}{2})=(-\frac{1}{2}, 3)$

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Example Question #22 : Midpoint Formula

Find the midpoint of a line segment that has endpoints at (9, 120) and (-24, 2) .

Possible Answers:

$(\frac{15}{2}, 61)$

$(-\frac{15}{2}, 61)$

$(-\frac{15}{2}, 59)$

$(-\frac{33}{2}, 60)$

Correct answer:

$(-\frac{15}{2}, 61)$

Explanation:

Recall how to find the midpoint of a line segment with endpoints at (x_1, y_1) and (x_2, y_2) :

$\text{Midpoint}=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

$\text{Midpoint}=(\frac{9+(-24)}{2}, \frac{120+2}{2})=(-\frac{15}{2}, 61)$

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Example Question #23 : Midpoint Formula

Find the midpoint of a line segment that has endpoints at (16, 7) and (8, 9) .

Possible Answers:

(12, 8)

(1, 8)

$(\frac{25}{2}, \frac{15}{2})$

(4, -1)

Correct answer:

(12, 8)

Explanation:

Recall how to find the midpoint of a line segment with endpoints at (x_1, y_1) and (x_2, y_2) :

$\text{Midpoint}=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

$\text{Midpoint}=(\frac{16+8}{2}, \frac{7+9}{2})=(12, 8)$

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Example Question #24 : Midpoint Formula

A line segment on the coordinate plane has its endpoints at (3.8, -1.7) and (9.2, 5.5) .

True or false: Its midpoint is located at (6.5, 3.6) .

Possible Answers:

True

False

Correct answer:

False

Explanation:

The midpoint of a line segment with endpoints $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is located at $\left ( \frac{x_{1}+x_{2}}{2} , \frac{y_{1}+ y_{2}}{2} \right )$ .

Set $x_{1}= 3.8, y_{1} =-1.7,x_{2} =9.2, y_{2}= 5.5$ , and evaluate both expressions:

$\frac{x_{1}+x_{2}}{2}= \frac{3.8+9.2}{2} = \frac{13}{2} = 6.5$

$\frac{-1.7+5.5}{2} =\frac{3.8}{2} = 1.9$

The midpoint is at (6.5, 1.9) , so the statement is false.

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

Find the midpoint between the points:

(0,4) and (2,8)

Possible Answers:

(1,6)

(1,7)

(0,0)

(1,1)

(2,4)

Correct answer:

(1,6)

Explanation:

The midpoint formula is

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

So for our two points the equation is

$\left(\frac{0+2}{2},\frac{4+8}{2}\right)$

$\left(\frac{2}{2},\frac{12}{2}\right)$

(1,6)

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

What is the midpoint between the two points:

(0,0) and (3,4)

Possible Answers:

(2,1.5)

(3,4)

(4,3)

(1.5,2)

(9,16)

Correct answer:

(1.5,2)

Explanation:

The midpoint formula is

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

If we plug in our points we get

$\left(\frac{0+3}{2},\frac{0+4}{2}\right)$

$\left(\frac{3}{2},\frac{4}{2}\right)$

(1.5,2)

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Example Question #21 : Lines

What is the midpoint between the two points:

(1,9) and (3, 5)

Possible Answers:

(2,5)

(2,7)

(7,2)

(1,9)

(3,4)

Correct answer:

(2,7)

Explanation:

The midpoint formula is

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

So lets plug in our two points, that gives us

$\left(\frac{1+3}{2},\frac{9+5}{2}\right)$

$\left(\frac{4}{2},\frac{14}{2}\right)$

(2,7)

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

What is the midpoint between the two points:

(0,0) and (4,5)

Possible Answers:

(2,2.5)

(2,5)

(4,2.5)

(4,5)

(2,2)

Correct answer:

(2,2.5)

Explanation:

The equation to find the midpoint between two points is

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

If we plug in our values of the two points we get

$\left(\frac{0+4}{2},\frac{0+5}{2}\right)$

$\left(\frac{4}{2},\frac{5}{2}\right)$

(2,2.5)

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

One of the endpoints of a line is $\small (8, -7)$ and the midpoint is $\small (1, -3)$ . What is the other endpoint?

Possible Answers:

$\small (15, 1 )$

$\small (-6, 1)$

$\small (-6, -11)$

$\small (15, -11)$

$\small (3, -1)$

Correct answer:

$\small (-6, 1)$

Explanation:

The midpoint's coordinates are the average of the endpoints'.

This means that the x-coordinates of the two endpoints have a mean of 1:

$\small \frac{8+x }{2} = 1$ multiply both sides by 2

$\small 8 + x = 2$ subtract 8

$\small x = -6$ this means the other endpoint's x-coordinate is -6

This also means that the y-coordinates of the two endpoints have a mean of -3:

$\small \frac{-7+y}{2} = -3$ multiply both sides by 2

$\small -7 + y = -6$ add 7 to both sides

$\small y = 1$

The coordinate pair that we're looking for is (-6, 1)

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Example Question #22 : Lines

The midpoint of a line is $\small (9, 9)$ and one of the endpoints is $\small (1, 17)$ . What is the other endpoint?

Possible Answers:

$\small (8, -8)$

$\small (-7, 1 )$

$\small (17, 25)$

$\small (-7, 25)$

$\small (17, 1)$

Correct answer:

$\small (17, 1)$

Explanation:

The midpoint's coordinates are just the mean of the endpoints'.

This means that the mean of the two x-coordinates is 9:

$\small \frac{x+ 1}{2} = 9$ multiply both sides by 2

$\small x + 1 = 18$ subtract 1

$\small x = 17$

This also means that the mean of the two y-coordinates is 9:

$\small \frac{y+17}{2} = 9$ multiply both sides by 2

$\small y + 17 = 18$ subtract 17

$\small y = 1$

So the other endpoint we were solving for is $\small (17, 1 )$

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Intermediate Geometry : Coordinate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #21 : Midpoint Formula

Example Question #22 : Midpoint Formula

Example Question #23 : Midpoint Formula

Example Question #24 : Midpoint Formula

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

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

Example Question #21 : Lines

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

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

Example Question #22 : Lines

All Intermediate Geometry Resources

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