How to find the endpoints of a line segment - Intermediate Geometry

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Question

Find the midpoint between the points:

and

Answer

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

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

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

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Question

What is the midpoint between the two points:

and

Answer

The midpoint formula is

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

If we plug in our points we get

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

$\left(\frac{3}{2},\frac{4}{2}\right)$

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Question

What is the midpoint between the two points:

and

Answer

The midpoint formula is

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

So lets plug in our two points, that gives us

$\left(\frac{1+3}{2},\frac{9+5}{2}\right)$

$\left(\frac{4}{2},\frac{14}{2}\right)$

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Question

What is the midpoint between the two points:

and

Answer

The equation to find the midpoint between two points is

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

If we plug in our values of the two points we get

$\left(\frac{0+4}{2},\frac{0+5}{2}\right)$

$\left(\frac{4}{2},\frac{5}{2}\right)$

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Question

One of the endpoints of a line is $\small (8, -7)$ and the midpoint is $\small (1, -3)$ . What is the other endpoint?

Answer

The midpoint's coordinates are the average of the endpoints'.

This means that the x-coordinates of the two endpoints have a mean of 1:

$\small \frac{8+x }{2} = 1$ multiply both sides by 2

$\small 8 + x = 2$ subtract 8

$\small x = -6$ this means the other endpoint's x-coordinate is -6

This also means that the y-coordinates of the two endpoints have a mean of -3:

$\small \frac{-7+y}{2} = -3$ multiply both sides by 2

$\small -7 + y = -6$ add 7 to both sides

$\small y = 1$

The coordinate pair that we're looking for is (-6, 1)

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Question

The midpoint of a line is $\small (9, 9)$ and one of the endpoints is $\small (1, 17)$ . What is the other endpoint?

Answer

The midpoint's coordinates are just the mean of the endpoints'.

This means that the mean of the two x-coordinates is 9:

$\small \frac{x+ 1}{2} = 9$ multiply both sides by 2

$\small x + 1 = 18$ subtract 1

$\small x = 17$

This also means that the mean of the two y-coordinates is 9:

$\small \frac{y+17}{2} = 9$ multiply both sides by 2

$\small y + 17 = 18$ subtract 17

$\small y = 1$

So the other endpoint we were solving for is $\small (17, 1 )$

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Question

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

Answer

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 .

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Question

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

Answer

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 .

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Question

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

Answer

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 .

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Question

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

Answer

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.

$\frac{-1+e}{2}=8$

Now, let's solve for the coordinate.

$\frac{-2+f}{2}=8$

The coordinate for the other endpoint must be .

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Question

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

Answer

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.

$\frac{6+g}{2}=7$

Now, let's solve for the coordinate.

$\frac{-1+h}{2}=2$

The coordinate for the other endpoint must be .

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Question

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

Answer

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.

$\frac{20+j}{2}=7$

Now, let's solve for the coordinate.

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

The coordinate for the other endpoint must be .

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Question

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

Answer

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 .

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Question

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

Answer

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.

$\frac{12+x}{2}=6$

Now, let's solve for the coordinate.

$\frac{-10+y}{2}=2$

The coordinate for the other endpoint must be .

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Question

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

Answer

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.

$\frac{-3+x}{2}=6$

Now, let's solve for the coordinate.

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

The coordinate for the other endpoint must be .

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Question

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

Answer

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 .

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Question

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

Answer

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 .

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Question

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

Answer

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.

$\frac{-8+x}{2}=3$

Now, let's solve for the coordinate.

$\frac{2+y}{2}=5$

The coordinate for the other endpoint must be .

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Question

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

Answer

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:

$\frac{-2+x_2}{2}=12$

Solve for .

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

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

The second endpoint must be at .

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Question

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

Answer

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 .

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