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Algebra 1 : Midpoint Formula

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Example Questions

Example Question #1 : Midpoint Formula

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

Possible Answers:

(22, –9)

(29, –15)

(8.5, –2)

(12, –11)

None of the other answers

Correct answer:

(29, –15)

Explanation:

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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Example Question #2 : Midpoint Formula

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

Possible Answers:

$\textup{1.5}$

$\textup{4}$

$\frac{1}{3}$

$\textup{None of the above.}$

$\frac{2}{3}$

Correct answer:

$\textup{None of the above.}$

Explanation:

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

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

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

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Example Question #3 : Midpoint Formula

Line segment AC has one endpoint at (-5,-2) . If this line's midpoint is at the origin, what are the coordinates of its other endpoint?

Possible Answers:

(5,2)

(-2,5)

(2,5)

(-2,5)

(5,-2)

Correct answer:

(5,2)

Explanation:

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 (-5,-2) 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 (5,2) . 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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Example Question #4 : Midpoint Formula

Line segment XY has a midpoint of (6,3) . If X is (2,8) what is Y?

Possible Answers:

(7,4)

(10,-2)

(10,2)

(2,6)

(4,5.5)

Correct answer:

(10,-2)

Explanation:

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 (2,8) as (x_1,y_1) . 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.

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

2+x_2=12

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

Doing similar arithmetic, y_2 will be solved to be .

Therefore, endpoint Y is (10,-2)

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Example Question #2 : Midpoint Formula

Line segment EF has a midpoint of (7,6) . If endpoint F is at (2,3) , what's the coordinate for endpoint E?

Possible Answers:

(12,7)

(12,9)

(10,12)

(-11,3)

(7,4)

Correct answer:

(12,9)

Explanation:

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 (2,3) as (x_1,y_1) . 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.

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

2+x_2=14

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

Doing similar arithmetic, y_2 will be solved to be .

Therefore, endpoint E is (12,9) .

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Example Question #3 : Midpoint Formula

Line segment DF has a midpoint of (4,8) . If endpoint D is at (22,7) , where is endpoint F?

Possible Answers:

(30,9)

(-14,9)

(5,-9)

(11,-18)

(22,13)

Correct answer:

(-14,9)

Explanation:

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 (22,7) as (x_1,y_1) . 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.

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

22+x_2=8

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

Doing similar arithmetic, y_2 will be solved to be .

Therefore, endpoint Y is (-14,9) .

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Example Question #7 : Midpoint Formula

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 (2,10) and the y-coordinate of the other endpoint is 5, find the value of the x-coordinate. To clarify, our endpoints are (2,10) and (?,5)

Possible Answers:

x=-1

x=3

x=0

x=2

x=1

Correct answer:

x=0

Explanation:

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:

2=2-x , which means our missing x-coordinate is 0. So, the endpoints of our line segment are $(0,5)\&(2,10)$ .

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

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

Possible Answers:

(4, 2)

(3, 8)

(4, 6)

(2, 6)

(7, 6)

Correct answer:

(7, 6)

Explanation:

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 (x_1, y_1) and (x_2, y_2) are the endpoints.

Now, here is what we know:

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

(x_1, y_1) = (-1, 2)

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 x_2 and y_2 to find the other endpoint.

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

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

6 = -1 + x_2

6 + 1 = -1 + x_2 + 1

7 = x_2

x_2 = 7

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

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

8 = 2 + y_2

8 - 2 = 2 + y_2 - 2

6 = y_2

y_2 = 6

This give us the point (7, 6) . Therefore, the other endpoint is (7, 6) .

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Example Question #1 : Midpoint Formula

A line segment has the midpoint (-2,3) . One end point of the line segment is located at (5,6) . What is the other end point?

Possible Answers:

(-9,0)

(-1,12)

(1,12)

(9,0)

Correct answer:

(-9,0)

Explanation:

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

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

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

(-4,6)=(5+x_2,6+y_2)

(x_2,y_2)=(-4-5,6-6)

(x_2,y_2)=(-9,0)

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Example Question #2 : Midpoint Formula

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

Possible Answers:

(–2, 2)

(3, 2)

(6, 4)

(2, 2)

(2, –2)

Correct answer:

(3, 2)

Explanation:

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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Algebra 1 : Midpoint Formula

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : Midpoint Formula

Example Question #2 : Midpoint Formula

Example Question #3 : Midpoint Formula

Example Question #4 : Midpoint Formula

Example Question #2 : Midpoint Formula

Example Question #3 : Midpoint Formula

Example Question #7 : Midpoint Formula

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

Example Question #1 : Midpoint Formula

Example Question #2 : Midpoint Formula

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