How to find the midpoint of a line segment - Intermediate Geometry

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Question

What is the midpoint between and ?

Answer

The midpoint is given by taking the mean, or average, of the and coordinates separately.

Let $P_{1}=\left ( 7,5 \right )$ and $P_{2} = \left (-1,3 \right )$

So the midpoint formula becomes $\frac{x_{1}\ +\ x_{2}}{2}= \frac{7-1}{2}=3$ and $\frac{y_{1}+y_{2}}{2}= \frac{-5 +3}{2}=-1$

So the midpoint is

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Question

Given two points $\left ( -3,-3 \right )$ and $\left ( -1,1 \right )$ and a line segment that connects the two.

What is the midpoint of the line segment? Addtionally, what is the length of the line segment?

Answer

The midpoint formula is as follows:

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

This makes sense; it is as simple as the average of the and components of each point.

For this problem, .

So the midpoint is $\left ( -2,-1 \right )$

The distance formula is as follows:

$distance=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The order you put the first and second components of each point does NOT matter, when a positive or negative number is squared, it will always come out positive.

So the distance is $\sqrt{(-3+1)^2+(-3-1)^2} =\sqrt{4+16}=\sqrt{20}=2\sqrt{5}$

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Question

What is the midpoint of a line segment connecting the points $\left ( -5,6 \right )$ and $\left ( -3,2 \right )\ ?$

Answer

Use the midpoint formula:

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

$(\frac{-5+(-3)}{2}, \frac{6+2}{2})=(\frac{-8}{2}, \frac{8}{2})=(-4,4)$

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Question

Find the midpoint of a line segment going from to .

Answer

To find the midpoint of a line segment, you must find the mean of the x values and the mean of the y values.

Our x values are and to find their mean, we do .

Our y values are and , so our mean is . Therefore, our midpoint must be .

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Question

Find the coordinate of the midpoint of the line segment connecting the pair of points

and .

Answer

The coordinate of the midpoint of the line segment connecting a pair of points is

$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

So for the pair of points and ,

we get:

$\left(\frac{2+4}{2}, \frac{2+12}{2}\right)=(3, 7)$

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Question

Find the coordinate of the midpoint of the line segment connecting the pair of points

and .

Answer

The coordinate of the midpoint of the line segment connecting a pair of points is

$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ .

So for the pair of points and

we get,

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

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Question

A line extends from the origin to $\small (14, -2)$ . What is its midpoint?

Answer

The origin is the point $\small (0,0)$ . We can use the regular midpoint formula with $\small (0,0)$ as $\small (x_{1}, y_{1})$ and the given point $\small (14, -2)$ as $\small (x_{2}, y_{2})$ :

$\small (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

$\small (\frac{0+14}{2} , \frac{0+-2}{2})$

$\small (\frac{14}{2}, \frac{-2}{2}) = (7, -1)$

Note that since the first point is the origin, we're averaging these points with 0, or in other words just dividing by 2.

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Question

There is a line segment between the two points (5, 10) and (3, 6). What formula will give you the correct mid-point of the line segment?

Answer

To find the mid-point given two points on the coordinate plane you are basically finding the average of the x-values and the average of the y-values. To do this follow the formula below:

$\Big( \frac{x_{1}+ x_{2}}{2} ; , \frac{y_{1}+y_{2}}{2} \Big )$

With our points this gives:

$\Big( \frac{5+3}{2} ; , \frac{10+6}{2} \Big )$

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Question

A line has endpoints at (4,5) and (-6,8). Where is its midpoint?

Answer

The midpoint is the point that is equidistant from each endpoint point.

We find it using a formula that computes the average of the x and y coordinates.

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

$\left(\frac{4+(-6)}{2},\frac{5+8}{2}\right)$

$\left(\frac{-2}{2},\frac{13}{2}\right)$

$\mathbf{\left(-1,\frac{13}{2}\right)}$

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Question

A line segment has the endpoints and . What is the midpoint of this line segment?

Answer

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.

$\text{Midpoint}=(\frac{2+1}{2}, \frac{4+8}{2})=(\frac{3}{2}, 6)$

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Question

A line segment has endpoints (8, 1) and (2, 5). What is the midpoint of this segment?

Answer

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.

$\text{Midpoint}=(\frac{8+2}{2}, \frac{1+5}{2})=(5, 3)$

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Question

A line segment has endpoints at (10, 2) and (-10, -6). What is the midpoint of this line?

Answer

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.

$\text{Midpoint}=(\frac{10-10}{2}, \frac{2-6}{2})=(0, -2)$

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Question

A line segments has endpoints at (-5, 1) and (-2, 9). What is the midpoint of this line segment?

Answer

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.

$\text{Midpoint}=(\frac{-5-2}{2}, \frac{1+9}{2})=(-\frac{7}{2}, 5)$

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Question

Find the midpoint of a line that has endpoints at (9, 2) and (11, 6).

Answer

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.

$\text{Midpoint}=(\frac{9+11}{2}, \frac{2+6}{2})=(10, 4)$

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Question

Find the midpoint of a line that has endpoints at (-15, 2) and (-2, 2).

Answer

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.

$\text{Midpoint}=(\frac{-15-2}{2}, \frac{2+2}{2})=(-\frac{17}{2}, 2)$

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Question

Find the midpoint of a line segment that has endpoints at (-9,-1) and (1,2).

Answer

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.

$\text{Midpoint}=(\frac{-9+1}{2}, \frac{-1+2}{2})=(-4, \frac{1}{2})$

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Question

Find the midpoint of a line segment with endpoints at (10, 2) and (-2, -8).

Answer

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.

$\text{Midpoint}=(\frac{10-2}{2}, \frac{2-8}{2})=(4, -3)$

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Question

Find the midpoint of a line segment that has endpoints at (5,1) and (-3, -10).

Answer

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.

$\text{Midpoint}=(\frac{5-3}{2}, \frac{1-10}{2})=(1, -\frac{9}{2})$

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Question

Find the midpoint of a line segment that has endpoints at (0.6, 9) and (-9.2, 7).

Answer

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.

$\text{Midpoint}=(\frac{0.6-9.2}{2}, \frac{9+7}{2})=(-4.3, 8)$

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Question

Find the midpoint of a line segment that has endpoints at (12, 12) and (-10, 24).

Answer

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.

$\text{Midpoint}=(\frac{12-10}{2}, \frac{12+24}{2})=(1,18)$

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