How to find the midpoint of a line segment - Algebra 1

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Question

What is the midpoint of the line segment between points and ?

Answer

$(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$ is the midpoint formula and will give the midpoint coordinates.

Plug in the numbers from the given ordered pairs:

$(\frac{1+(-7)}{2},\frac{8+10}{2})$

$(\frac{-6}{2},\frac{18}{2})$

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Question

Find the midpoint on the line segment from (2, 3) to (4, 1).

Answer

By using the midpoint formula, we can find the x and y coodinantes fo the midpoint.

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

$\frac{x_1+x_2}{2}=\frac{2+4}{2}=\frac{6}{2}=3$

$\frac{y_1+y_2}{2}=\frac{3+1}{2}=\frac{4}{2}=2$

Our coordinates are (3, 2).

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Question

A line is drawn from (2,4) to (8,28). What are the coordinates of its midpoint?

Answer

The length to the midpoint is the difference between the two points divided by two. That number must then be added to the point:

$X_{midpoint}=X_{1}+\frac{\left (X_{2}-X_{1} \right )}{2}=2+\frac{\left (8-2 \right )}{2}=2+\frac{6}{2}=2+3=5$

$Y_{midpoint}=Y_{1}+\frac{\left (Y_{2}-Y_{1} \right )}{2}=4+\frac{\left (28-4 \right )}{2}=4+\frac{24}{2}=4+12=16$

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Question

Point X (2, 9) and Point Y (8, 3) are endpoints on a line segment. What is the Midpoint M of that line segment?

Answer

To find the midpoint of a line segment, you add together the components and divide by two ( $\frac{2+8}{2}$ = 5) , do the same for ( $\frac{3+9}{2}$ =6). The answer is (5, 6).

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Question

Find the midpoint that falls between and .

Answer

The midpoint formula is $\left ( \frac{x_{1} + x_{2}}{}2, \frac{y_{1} + y_{2}}{2}\right)$ .

When we plug in our points, we get $\left ( \frac{14}{2}, \frac{-4}{2}\right)$ .

So, our final answer is .

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Question

What is the midpoint of the points (3,12) and (9,15)?

Answer

To find the midpoint we must know the midpoint formula which is

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

We then take the -coordinate from the first point and plug it into the formula as $x_{1}$ .

We take the -coordinate from the second point and plug it into the formula as $x_{2}$ .

We then do the same for $y_{1}$ and $y_{2}$ .

With all of the points plugged in our equation will look like this.

$(\frac{3+9}{2},\frac{12+15}{2})$

We then perform the necessary addition and division to get the answer of

$(\frac{12}{2},\frac{27}{2})=(6,13.5)$

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Question

A line segment begins at and ends at the point . What is the location of its midpoint?

Answer

The difference in -values is 14 and the difference in -values is 8. The midpoint therefore differs by values of 7 and 4 from either of the endpoints.

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Question

Find the midpoint of the line segment that connects the two points below.

Point 1:

Point 2:

Answer

The average of the the -coordinates and the average of the y-coordinates of the given points will give you the mid-point of the line that connects the points.

$\textup{midpoint}=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$ , where is and is .

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

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

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

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Question

$\textup{Two points }S\left ( 5,2\right )\textup{ and }T\left ( -3,10\right )\textup{ lie in the }xy\textup{-coordinate plane.}$

$\textup{What is the midpoint of line segment }\overline{ST}\textup{ ?}$

Answer

$\textup{The coordinates of the midpoint are the averages of the other points' coordinates.}$

$x\textup{-coordinate}=\frac{5+\left ( -3)\right )}{2}=1;;;;y\textup{-coordinate}=\frac{2+10}{2}=6$

$\textup{The midpoint is at }\left (1,6\right )$

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Question

A line has endpoints of and . What is its midpoint?

Answer

The midpoint formula is $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$

To find the midpoint of and , you simply plug in the points into the midpoint formula: $(\frac{5+-1}{2},\frac{-2+0}{2})$ , which gives you the point .

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Question

A line has endpoints of $\left ( 10,-2 \right )$ and $\left ( -4,8 \right )$ . What is its midpoint?

Answer

You can find the midpoint of the line by using the midpoint formula: $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$ . Plug the endpoints into the formula to get $(\frac{10+-4}{2},\frac{{-2}+8}{2})$ , or .

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Question

What is the midpoint of a line with endpoints of and ?

Answer

To find the midpoint, use the midpoint formula: $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$ . Plug in the two ordered pairs to the formula to get: $(\frac{-10+4}{2},\frac{14+2}{2})$ . Doing this will give you a solution of .

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Question

What is the midpoint of a line with endpoints of and ?

Answer

To find the midpoint, you can use the midpoint formula: $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$ .

Plug in and into the formula: $(\frac{9+-1}{2},\frac{-3+5}{2})$ to get .

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Question

A line segment on the coordinate plane has endpoints and . Which quadrant or axis contains its midpoint?

Answer

The -coordinate of the midpoint is

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

which is negative.

The -coordinate of the midpoint is

$\frac{-4+10}{2} =3$ ,

which is positive.

Since the midpoint has a negative -coordinate and a positive -coordinate, the midpoint is in Quadrant II.

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Question

A line segment on the coordinate plane has endpoints and . Which quadrant or axis contains its midpoint?

Answer

The -coordinate of the midpoint is

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

which is positive.

The -coordinate of the midpoint is

$\frac{-4+7}{2} = \frac{3}{2}$ ,

which is positive.

Since both coordinates are positive, the midpoint is in Quadrant I.

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Question

Determine the midpoint between the points and

Answer

To find the midpoint you are actually finding the average of the two values and the average of the two values.

Midpoint formula: $(\frac{x_{1}+x_{2}}{2}), (\frac{y_{1}+y_{2}}{2})$

, so we plug our points in to the equation,

$(\frac{-1+(-7)}{2}, \frac{5+3}{2})$

Simplify and divide

$(\frac{-8}{2},\frac{8}{2})$

Midpoint:

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Question

Find the midpoint of the line segment with endpoints and .

Answer

Use the midpoint formula:

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

Substitute:

$\frac{x_{1}+x_{2}}{2} =\frac{3.6+7.2}{2} = \frac{10.8}{2} = 5.4$

$\frac{y_{1}+y_{2}}{2}} \right = \frac{9.1+(-1.5)}{2}} = \frac{7.6}{2} = 3.8$

The midpoint is

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Question

Find the midpoint of the line segment with endpoints and .

Answer

Use the midpoint formula:

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

Substitute:

$\frac{x_{1}+x_{2}}{2} =\frac{1.6+(-1.4)}{2} = \frac{0.2}{2} = 0.1$

$\frac{y_{1}+y_{2}}{2}} \right = \frac{7.3+4.7)}{2}} = \frac{12}{2} = 6$

The midpoint is

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Question

Find the midpoint of the line segment with endpoints $\left ( \frac{1}{3}, \frac{5}{6} \right )$ and $\left ( -\frac{5}{6},\frac{2}{3} \right )$ .

Answer

Use the midpoint formula:

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

Substitute:

$\frac{x_{1}+x_{2}}{2} = \frac{ \frac{1}{3} + \left (-\frac{5}{6} \right )}{2}= \frac{ \frac{2}{6} - \frac{5}{6} }{2}= \frac{-\frac{3}{6}}{2}= \frac{-\frac{1}{2}}{2} = -\frac{1}{2} \div 2 = -\frac{1}{2} \cdot\frac{1}{2} =-\frac{1}{4}$

$\frac{y_{1}+y_{2}}{2} = \frac{ \frac{5}{6} + \frac{2}{3}}{2} = \frac{ \frac{5}{6} + \frac{4}{6}}{2} = \frac{ \frac{9}{6} }{2}= \frac{ \frac{3}{2} }{2} = \frac{3}{2} \div 2 = \frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}$

The midpoint is $\left ( -\frac{1 }{4}, \frac{3 }{4} \right )$ .

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Question

What is the midpoint of a line with endpoints (2,5) and (18,-9)?

Answer

To find the midpoint of the line, plug the endpoints into the distance formula: $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$ . This will give you $(\frac{2+18}{2},\frac{5+-9}{2})$ , or a midpoint of .

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