Midpoint Formula - Algebra 1

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Question

Point A is (5, 7). Point B is (x, y). The midpoint of AB is (17, –4). What is the value of B?

Answer

Point A is (5, 7). Point B is (x, y). The midpoint of AB is (17, –4). What is the value of B?

We need to use our generalized midpoint formula:

MP = ( (5 + x)/2, (7 + y)/2 )

Solve each separately:

(5 + x)/2 = 17 → 5 + x = 34 → x = 29

(7 + y)/2 = –4 → 7 + y = –8 → y = –15

Therefore, B is (29, –15).

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Question

A line segment has the midpoint . One end point of the line segment is located at . What is the other end point?

Answer

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

$(-2,3)=(\frac{5+x_2}{2},\frac{6+y_2}{2})$

$2(-2,3)=2(\frac{5+x_2}{2},\frac{6+y_2}{2})$

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Question

$\textup{Find the distance between the midpoint of the line segment}\ \textup{ between points } (-1,-1)\textup{ and } (2,3) \textup{ and the midpoint of the line}\\textup{ segment between points }(-1,-1)\textup{ and (2,-1).}$

Answer

$\textup{The midpoint of the first line segment has coordinates }\\left(\frac{-1+2}{2},\frac{-1+3}{2} \right )=\left(0.5,1 \right) \textup{and the midpoint of the second}\\textup{ line segment has coordinates }\left(\frac{-1+2}{2},\frac{-1-1}{2} \right )=\left(0.5,-1 \right ).$

$\textup{The segment that joins the two midpoints is on a vertical line, }\\textup{therefore its length is the absolute difference between the two}\\textup{midpoints' }y\textup{-coordinates }\left | 1-(-1)\right |=2.$

$\textup{Alternatively, one can draw all the points and segments and find}\\textup{the distance by exploiting the similarity of the two right triangles.}$

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Question

Line segment AC has one endpoint at . If this line's midpoint is at the origin, what are the coordinates of its other endpoint?

Answer

A line's midpoint is the coordinate pair of that line which has the same number of points on either side of it. It bisects the line in two equal parts.

Solution:

We are given that the line has an endpoint at and its midpoint is on the origin. This known point would be in the Quadrant III and since on the opposite side of the midpoint there is exactly as much line we know that the other half of our line will lie in the Quadrant I. Add the absolute value of our known point to the coordinates of the origin to get . This is the unknown endpoint. You should recognize that this end point is exactly the same distance in the x and y direction (just opposite) as our given endpoint.

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Question

Line segment XY has a midpoint of . If X is what is Y?

Answer

For this kind of problem, it's important to keep in mind how midpoint is solved for:

$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$ where is the midpoint coordinate.

Because we've been given the midpoint already, we'll have to solve this problem backwards. Since we've been given one of the endpoints (X) and we just need to solve for the other end point (Y), we may arbitrarily assign as . If we substitute in the two coordinates - an endpoint and the midpoint - we should be able to solve for the unknown endpoint.

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

$(\frac{2+x_2}{2},\frac{8+y_2}{2}) = (6,3)$

It may be visually easier to break the arithmetic into separate operations.

$\frac{2+x_2}{2}=6$ and $\frac{8+y_2}{2}=3$

By separating the x and y components, we can easily solve for the missing endpoint now.

$2[\frac{2+x_2}{2}=6]$

$x_2={\color{Blue} 10}$

Doing similar arithmetic, will be solved to be .

Therefore, endpoint Y is

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Question

Line segment DF has a midpoint of . If endpoint D is at , where is endpoint F?

Answer

For this kind of problem, it's important to keep in mind how midpoint is solved for:

$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$ where is the midpoint coordinate.

Because we've been given the midpoint already, we'll have to solve this problem backwards. Since we've been given one of the endpoints (D) and we just need to solve for the other end point (F), we may arbitrarily assign as . If we substitute in the two coordinates - an endpoint and the midpoint - we should be able to solve for the unknown endpoint.

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

$(\frac{22+x_2}{2},\frac{7+y_2}{2}) = (4,8)$

It may be visually easier to break the arithmetic into separate operations.

$\frac{22+x_2}{2}=4$ and $\frac{7+y_2}{2}=8$

By separating the x and y components, we can easily solve for the missing endpoint now.

$2[\frac{22+x_2}{2}=4]$

$x_2={\color{Blue} -14}$

Doing similar arithmetic, will be solved to be .

Therefore, endpoint Y is .

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Question

Line segment EF has a midpoint of . If endpoint F is at , what's the coordinate for endpoint E?

Answer

For this kind of problem, it's important to keep in mind how midpoint is solved for:

$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$ where is the midpoint coordinate.

Because we've been given the midpoint already, we'll have to solve this problem backwards. Since we've been given one of the endpoints (F) and we just need to solve for the other end point (E), we may arbitrarily assign as . If we substitute in the two coordinates - an endpoint and the midpoint - we should be able to solve for the unknown endpoint.

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

$(\frac{2+x_2}{2},\frac{3+y_2}{2}) = (7,6)$

It may be visually easier to break the arithmetic into separate operations.

$\frac{2+x_2}{2}=7$ and $\frac{3+y_2}{2}=6$

By separating the x and y components, we can easily solve for the missing endpoint now.

$2[\frac{2+x_2}{2}=7]$

$x_2={\color{Blue} 12}$

Doing similar arithmetic, will be solved to be .

Therefore, endpoint E is .

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Question

The midpoint of a line segment is represented by the point $\left(\frac{15}{2},1\right)$ . If the coordinates for one of its endpoints are and the y-coordinate of the other endpoint is 5, find the value of the x-coordinate. To clarify, our endpoints are and

Answer

We know that the midpoint of our line segment is $\left(\frac{15}{2},1\right)$ . To find the x-coordinate of this segment, we work backwards, starting with our midpoint formula. In this case, we only need to use the midpoint formula to solve for the x-coordinate, which looks like:

$1=\frac{2-x}{2}$

Next, multiply both sides of the equation by 2, which gives us:

, which means our missing x-coordinate is 0. So, the endpoints of our line segment are .

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Question

If the midpoint of a line segment is (3, 4) and one endpoint is (-1, 2), find the other endpoint.

Answer

To solve, we will using the midpoint formula and substitute what we know. The midpoint formula is:

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

where and are the endpoints.

Now, here is what we know:

$\text{midpoint} = (3, 4)$

Here is what we are solving for

$(x_2, y_2) = \ ???$

So, we will substitute. We get

$(3, 4) = (\frac{-1 + x_2}{2}, \frac{2 + y_2}{2})$

We can divide this into parts. We know

$3 = \frac{-1 + x_2}{2}$

and

$4 = \frac{2 + y_2}{2}$

So, we can solve for and to find the other endpoint.

$3 = \frac{-1 + x_2}{2}$

$3 \cdot 2 = \frac{-1 + x_2}{2} \cdot 2$

$4 = \frac{2 + y_2}{2}$

$4 \cdot 2 = \frac{2 + y_2}{2} \cdot 2$

This give us the point . Therefore, the other endpoint is .

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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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