How to find the endpoints of a line segment

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Math › How to find the endpoints of a line segment

Questions 1 - 10

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

$\textup{None of the above.}$

$\frac{1}{3}$

$\frac{2}{3}$

$\textup{1.5}$

$\textup{4}$

Explanation

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

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

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

A line segment has an endpoint at and a midpoint at . Find the coordinates of the other endpoint.

Explanation

Recall how to find the midpoint of a line segment:

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

where $(x_1, y_1)\text{ and}(y_1, y_2)$ are the endpoints.

Let's first focus on the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

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

Solve for .

Next, find the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{1+y_2}{2}=-10$

The second endpoint must be at .

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

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 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

A line segment has an endpoint at and a midpoint at . Find the coordinates of the other endpoint.

Explanation

Recall how to find the midpoint of a line segment:

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

where $(x_1, y_1)\text{ and}(y_1, y_2)$ are the endpoints.

Let's first focus on the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{11+x_2}{2}=5$

Solve for .

Next, find the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{-2+y_2}{2}=0$

The second endpoint must be at .

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

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 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 .

A line segment on the coordinate plane has an endpoint at ; its midpoint is at .

True or false: Its other endpoint is located at .

True

False

Explanation

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

Therefore, set

$\frac{x_{1}+x_{2}}{2} = 9.2$ and $\frac{y_{1}+ y_{2}}{2} = 5.5$

In the first equation, set $x_{2} = 3.8$ and solve for $x_{1}$ :

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

Multiply both sides by 2:

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

$x_{1}+3.8= 18.4$

Subtract 3.8 from both sides:

$x_{1}+3.8- 3.8 = 18.4 - 3.8$

$x_{1} = 14.6$

In the second equation, set $y_{2} = -1.7$ and solve for $x_{2}$ :

$\frac{y_{1}+ (-1.7)}{2} = 5.5$

Multiply both sides by 2:

$\frac{y_{1} -1.7}{2} \cdot 2 = 5.5 \cdot 2$

$y_{1} -1.7 = 11$

Add 1.7 to both sides:

$y_{1} -1.7+ 1.7 = 11 + 1.7$

$y_{1} = 12.7$

The other endpoint is indeed at , so the statement is true.

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

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 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 .

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

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 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.