Geometry - How to find the endpoints of a line segment

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 .

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 .

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.

A line with endpoints and has a midpoint of . What are the coordinates of the unknown endpoint?

Explanation

Recall how to find the midpoint of a line:

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

In other words, the coordinates of the midpoint are just the average of the and coordinates of the endpoints.

We can then solve for the coordinate of the second endpoint.

Let's start with the coordinate by using the given midpoint and endpoint.

$\frac{5+l}{2}=-1$

Now, let's solve for the coordinate.

$\frac{m+5}{2}=1$

The coordinate for the other endpoint must be .

Find the midpoint between the points:

and

Explanation

The midpoint formula is

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

So for our two points the equation is

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

$\left(\frac{2}{2},\frac{12}{2}\right)$

The midpoint for a line with an endpoint at is . What are the coordinates for the other endpoint?

Explanation

Recall how to find the midpoint of a line:

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

In other words, the coordinates of the midpoint are just the average of the and coordinates of the endpoints.

We can then solve for the coordinate of the second endpoint.

Let's start with the coordinate by using the given midpoint and endpoint.

$\frac{3+x}{2}=7$

Now, let's solve for the coordinate.

$\frac{y+1}{2}=8$

The coordinate for the other endpoint must be .

A line with endpoints of and has a midpoint of . What are the coordinates of the unknown endpoint?

Explanation

Recall how to find the midpoint of a line:

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

In other words, the coordinates of the midpoint are just the average of the and coordinates of the endpoints.

We can then solve for the coordinate of the second endpoint.

Let's start with the coordinate by using the given midpoint and endpoint.

$\frac{4+x}{2}=8$

Now, let's solve for the coordinate.

$\frac{-4+y}{2}=8$

The coordinate for the other endpoint must be .

A line with the endpoints and has a midpoint at . What are the coordinates for the unknown endpoint?

Explanation

Recall how to find the midpoint of a line:

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

In other words, the coordinates of the midpoint are just the average of the and coordinates of the endpoints.

We can then solve for the coordinate of the second endpoint.

Let's start with the coordinate by using the given midpoint and endpoint.

$\frac{5+a}{2}=-10$

Now, let's solve for the coordinate.

$\frac{b+1}{2}=2$

The coordinate for the other endpoint must be .

A line with endpoints of and has a midpoint of . What are the coordinates of the unknown endpoint?

Explanation

Recall how to find the midpoint of a line:

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

In other words, the coordinates of the midpoint are just the average of the and coordinates of the endpoints.

We can then solve for the coordinate of the second endpoint.

Let's start with the coordinate by using the given midpoint and endpoint.

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

Now, let's solve for the coordinate.

$\frac{-9+y}{2}=6$

The coordinate for the other endpoint must be .

A line with endpoints at and has a midpoint of . What are the coordinates for the unknown endpoint?

Explanation

Recall how to find the midpoint of a line:

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

In other words, the coordinates of the midpoint are just the average of the and coordinates of the endpoints.

We can then solve for the coordinate of the second endpoint.

Let's start with the coordinate by using the given midpoint and endpoint.

$\frac{-9+c}{2}=5$

Now, let's solve for the coordinate.

$\frac{-2+d}{2}=-9$

The coordinate for the other endpoint must be .

How to find the endpoints of a line segment

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