# How to find the second endpoint algebraically when given one endpoint and the midpoint

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, volunteer authors worked to edit and improve it over time.

This wikiHow article teaches you how to find the endpoint of a line segment when you’re given another endpoint and the midpoint. The formula is as follows: ( x 3 , y 3 ) = ( x 1 + x 2 2 , y 1 + y 2 2 ) <\displaystyle (x_<3>,y_<3>)=(<\frac +x_<2>><2>>,<\frac +y_<2>><2>>)> . In this article, the midpoint ( x 3 , y 3 <\displaystyle x_<3>,y_<3>> ) = <\displaystyle => ( − 6 , − 2 ) <\displaystyle (-6,-2)> and the given endpoint ( x 1 , y 1 <\displaystyle x_<1>,y_<1>> ) = <\displaystyle => ( − 3 , − 5 ) <\displaystyle (-3,-5)> are used as examples. The rest is algebra: isolate x 2 <\displaystyle x_<2>> and y 2 <\displaystyle y_<2>> (separately) to find the second endpoint coordinates. See a solution process below:

#### Explanation:

The formula to find the mid-point of a line segment give the two end points is:

#M = ((color(red)(x_1) + color(blue)(x_2))/2 , (color(red)(y_1) + color(blue)(y_2))/2)#

Where #M# is the midpoint and the given points are:

#(color(red)(x_1, y_1))# and #(color(blue)(x_2, y_2))#

#(-2, 9) = ((color(red)(2) + color(blue)(x_2))/2 , (color(red)(8) + color(blue)(y_2))/2)#

Solving for #x# gives:

#-2 = (color(red)(2) + color(blue)(x_2))/2#

#color(green)(2) xx -2 = color(green)(2) xx (color(red)(2) + color(blue)(x_2))/2#

#-4 = cancel(color(green)(2)) xx (color(red)(2) + color(blue)(x_2))/color(green)(cancel(color(black)(2)))#

#-4 = color(red)(2) + color(blue)(x_2)#

#-2 – 4 = -2 + color(red)(2) + color(blue)(x_2)#

#-6 = 0 + color(blue)(x_2)#

Solving for #y# gives:

#9 = (color(red)(8) + color(blue)(y_2))/2#

#color(green)(2) xx 9 = color(green)(2) xx (color(red)(8) + color(blue)(y_2))/2#

#18 = cancel(color(green)(2)) xx (color(red)(8) + color(blue)(y_2))/color(green)(cancel(color(black)(2)))#

#18 = color(red)(8) + color(blue)(y_2)#

#-8 + 18 = -8 + color(red)(8) + color(blue)(y_2)#

#10 = 0 + color(blue)(y_2)#

The other end point is: #(color(blue)(-6, 10))# The coordinate of the other endpoint is #(-6, 10)#

#### Explanation: This is a simple number line from #0# to #10# . Now, I ask you, what is an easy way to calculate the midpoint between #0# and #10# ? Of course it’s #5# but is there a formula we can derive to calculate the midpoint between any two points? Well, let’s see. If we took #0# and #10# , added them together, then divided the quotient by #2# , boom! We get #5# !

Let’s try it with two more numbers. How about #2# and #4# ? If we were to add the two up, then divide the result by #2# , we’d get #3# – also the midpoint between #2# and #4# .

Now we can see that there’s a formula that we can use to calculate the midpoint between any two numbers! Add the two end numbers up and divide the result by #2# ! This formula – known as the Midpoint Formula – is shown below:

For any two endpoints #x_1# and #x_2# on the number line,

So how does all of this relate to the problem? Well, remember that the graph is a coordinate plane, made up of two axes – the x-axis and the y-axis. And you can think of each axis to be a number line!

So now, our task is to derive a formula for finding the midpoint between any two points on the coordinate point. That way, we can write a relationship between the two endpoints on the coordinate plane and the midpoint between those two points.

Suppose there was a graph like the one shown below: Suppose point A had the coordinates #(x_1, y_1)# and point B had the coordinates #(x_2, y_2)# . And suppose there was also a point M that was the midpoint between points A and B. Now, we want to write the midpoint’s coordinates in terms of the #x# and #y# coordinates of points A and B. That way, we can connect the coordinates of points A and B with the coordinates of their midpoint. Now, the coordinates of the midpoint are essentially the midpoint between the two x-coordinates and the midpoint between the two y-coordinates.

Let’s focus on the x-axis first and treat it as a number line, except without numbers. On the x-axis, we have the x-coordinate of point A as well as the x-coordinate of point B. From our earlier rule, we can state that the midpoint between the two x-coordinates is:

Likewise, the midpoint between the two y-coordinates is:

As we stated earlier, the coordinates of the midpoint between points A and B are the midpoint between the two x-coordinates and the midpoint between the two y-coordinates. Combining the two statements above, we get the conclusion that the coordinates of the midpoint between points A and B are:

Now, with that information, we can substitute in the values mentioned in the question. We let point A and its coordinates be our missing endpoint, point B and its coordinates be our known endpoint, and point M, our midpoint, and its coordinates be the midpoint, like so:

This equation tells us that #-2=(x_1+2)/2# and that #9=(y_1+8)/2# . Next, we start solving the equations to figure out what #x_1# and #y_1# are.

And now, let’s solve the right equation:

And now, we put the two values together to form the answer: ### Examples

#### Lessons

1. Determine the midpoint of the line segment with the given endpoints.
1. A ( 3 , 7 ) , B ( 9 , 1 ) A(3,7), B(9,1) A ( 3 , 7 ) , B ( 9 , 1 )
2. A ( x + 3 , y − 2 ) , B ( x − 2 , y + 9 ) A(x+3,y-2), B(x-2,y+9) A ( x + 3 , y − 2 ) , B ( x − 2 , y + 9 )
2. Determine the missing x.
1. A ( 3 , 7 ) , B ( 9 , x ) A(3,7), B(9,x) A ( 3 , 7 ) , B ( 9 , x ) ; Midpoint ( 6 , 4 ) (6,4) ( 6 , 4 )
2. A ( x , 2 ) , B ( 3 x , 18 ) A(x,2), B(3x,18) A ( x , 2 ) , B ( 3 x , 18 ) ; Midpoint ( − 8 , 10 ) (-8,10) ( − 8 , 10 )

## What is the midpoint formula?

At times, you may need to find the midpoint between two points. This usually comes into play when a question asks you to divide up a line into two equal halves, or in word problems when it asks you to find the midpoint.

If you think about it, the idea of finding the number that lies between a set of numbers is quite easy. What do you usually do? You’ll take the average of them. First add them together, then divide it by two. The midpoint formula is similar and instead of taking the average of just a number, you’ll have to take the average of the x- and y- values from two points separately.

The midpoint of the x-value is halfway between the two points’ x-values. The midpoint of the y-value is halfway between the two points’ y-values. This makes a lot of sense, right?

Of course, this means that you’ll need to know the coordinates of the two points in question before you can find its midpoint. But once you know the endpoints of a line segment, you can easily find the midpoint.

This is all summed up nicely with the midpoint formula, which is as follows:

## How to find midpoint

In order to see the midpoint formula in use, let’s look at an example.

Determine the midpoint of the line segment with the given endpoints.

We can use the formula for midpoint to determine the midpoint: M = ( x 1 + x 2 2 , y 1 + y 2 2 ) M=(\frac<2>, \frac<2>) M = ( 2 x 1 ​ + x 2 ​ ​ , 2 y 1 ​ + y 2 ​ ​ )

First, plug the x and y value into the formula:

Then, we can calculate the midpoint:

Let’s take a closer look at the steps from the above example. Firstly, we are given the coordinates of the two points for which we must find the midpoint of a segment. This is all you’ll need in order to make use of the midpoint formula.

First, we’ll take out the x-coordinates first so that we can work on finding the midpoint’s x-coordinate. We have 3 and 9 taken from point A and B respectively. Simply add the two together then divide it by 2 to get the average, a.k.a the midpoint for the x-value.

Then, do the same thing with the y-coordinates. We have 7 and 1 from, again, point A and B respectively. Adding that together gives us 8, and dividing it by two gives us 4.

We have determined that the midpoint between two points lies exactly at the coordinates (6,4) through the midpoint rule using the midpoint equation. If you plotted the line segment out onto a graph, and then dropped a point down onto (6,4), you’ll indeed see that this is the point that separates the segment into two equal portions.

Another word for a point that cuts a line into two equal segments is called a bisector. Some questions may ask you to find the bisector of a line, which is basically asking you for a midpoint. You may also come across questions asking if a certain coordinate is a bisector, and you’ll have to determine with the midpoint formula whether you get the midpoints that was stated. If not, then it isn’t a bisector.

Check if you got the right answers by referencing this midpoint calculator if you get stuck.

Before we actually delve and learn how to apply or use the midpoint formula to solve problems, let’s pause for a moment and have a practical understanding of it. Think of the midpoint as the “halfway” or middle point of a line segment. This so-called center point divides the line segment into two equal or congruent parts. NOTE: The midpoint of line segment AC denoted by the symbol \overline is located at point B . This implies that line segment AB , written as \overline , and line segment BC , written also as \overline , have equal measure. Therefore, \overline = \overline .

In a nutshell, the formula to find the midpoint of two given points is the following.

## The Midpoint Formula The midpoint M of the line segment with endpoints A ( x 1, y 1) and B ( x 2, y 2) is calculated as follows: Observations:

a) The x-coordinate of the midpoint is the average of the x -values from the given points.

b) The y-coordinate of the midpoint is the average of the y -values from the given points.

### Examples on How to Use the Midpoint Formula

Let’s go over five (5) different examples to see the midpoint formula in action!

Example 1: Find the midpoint of the line segment joined by the endpoints (–3, 3) and (5, 3).

When you plot the points in the xy-axis and join them with a ruler, the line segment is obviously horizontal because the y-coordinates of points are equal. In doing so, it is easy to approximate or guess the midpoint even without the midpoint formula. You can do just that by counting the same number of units from both sides of the endpoints until it reaches the center.

However, let’s work it out using the formula to find the midpoint.

Let (–3, 3) be the first point so x 1 = –3 and y 1 = 3. In the same manner if (5,3) is the second point then x 2 = 5 and y 2 = 3. Substitute these values in the formula, and simplify to get the midpoint. Here are the points plotted in the Cartesian plane, together with the calculated value of midpoint. Example 2: Find the center point of the line segment joined by the endpoints (1, 5) and (1, –1) using the midpoint formula.

This particular line segment is clearly vertical because the two points have the same x-coordinates. More so, plotting the points in the xy-axis verifies the case. Like in our previous example, the midpoint of this vertical line segment can easily be approximated by counting the same number of units from both sides of the endpoints.

Anyway, let’s solve this using the formula. Here’s how it looks on the graph. Example 3: Find the midpoint of the line segment joined by the endpoints (–4, 5) and (2, –3).

Notice that when you plot the line segment generated by the given endpoints, the resulting line segment is neither horizontal nor vertical, unlike the last two examples. You instead get a diagonal line segment. This time around, it is more difficult to guess or approximate the midpoint. However, with the use of the formula, this should not be a problem.

Now, substitute and evaluate the values in the midpoint formula. Here’s the graph. Example 4: Find the missing value of h in the points (5, 7) and (1, h ) if its midpoint is at (3, –2).

Since the midpoint is actually given to us, start by setting the formula equal to the numerical value of the midpoint. Just like this…

We let ( x 1 , y 1 ) = ( 5, 7) and ( x 2 , y 2 ) = (1, h ). Then substitute these values in the formula. If you observe, two points are equal whenever their corresponding coordinates are the same. That is, the x -values are equal, and the y -values are equal as well.

Notice, the x -values are both equal to 3. Great!

But we want to do the same thing with the y -coordinates by setting them equal to each other. By doing so, we created a simple equation which we can be solved for the missing value of h . Example 5: Find the center of a circle whose diameter has endpoints (–1, –5) and (5, –1).

If you think about it, the center of a circle is just the midpoint of the diameter. This problem is simply reduced to solving the midpoint of the line segment with endpoints (− 1, − 5) and (5, −1).

I am sure that you already know how to do this. Just a word of caution though, be very careful when adding or subtracting numbers with the same or different signs. This is where silly mistakes happen because students tend to “relax” when performing basic arithmetic operations. So don’t be. Here’s the graph of the circle showing its diameter and center.

## Purplemath

Sometimes you need to find the point that is exactly midway between two other points. For instance, you might need to find a line that bisects (divides into two equal halves) a given line segment. This middle point is called the “midpoint”. The concept doesn’t come up often, but the Formula is quite simple and obvious, so you should easily be able to remember it for later.

Think about it this way: If you are given two numbers, you can find the number exactly between them by averaging them, by adding them together and dividing by two. For example, the number exactly halfway between 5 and 10 is:

Content Continues Below

## MathHelp.com The Midpoint Formula works exactly the same way. If you need to find the point that is exactly halfway between two given points, just average the x -values and the y -values.

#### Find the midpoint P between (−1, 2) and (3, −6) .

First, I apply the Midpoint Formula; then, I’ll simplify:

So the answer is P = (1, −2) .

Technically, the Midpoint Formula is the following:

The Midpoint Formula: The midpoint of two points, (x1, y1) and (x2, y2) is the point M found by the following formula:

But as long as you remember that you’re averaging the two points’ x – and y -values, you’ll do fine. It won’t matter which point you pick to be the “first” point you plug in. Just make sure that you’re adding an x to an x , and a y to a y .

#### Find the midpoint P between (6.4, 3) and (−10.7, 4) .

I’ll apply the Midpoint Formula, and simplify:

So the answer is P = (−2.15, 3.5) .

Content Continues Below

#### Find the value of p so that (−2, 2.5) is the midpoint between (p, 2) and (−1, 3) .

I’ll apply the Midpoint Formula:

The y -coordinates already match. This reduces the problem to needing to compare the x -coordinates, “equating” them (that is, setting them equal, because they must be the same) and solving the resulting equation to figure out what p is. This will give me the value necessary for making the x -values match. So: If you have the coordinates (x, y) of the endpoints of a line segment, finding the midpoint of the segment is very simple. Just take the average of the two x-coordinates of the endpoints (add them and divide by two) to get the x-coordinate of the midpoint. Then do the same with the y-coordinates.

The midpoint “formula” is given below. It implies that the x coordinate of the midpoint is half way between the x coordinates of the endpoints, and likewise for the y-coordinates. This is proven using triangle congruence below the formula.

#### Midpoint formula

The coordinates of the midpoint of a line segment on a plane (2-dimensional) are:

### Proof of the midpoint formula

If we take line segment AE (right →) with midpoint C, we can draw all of the dashed lines parallel to the coordinate axes. All vertical lines are || and all horizontal lines are ||. Now because parallel lines yield congruent pairs of corresponding angles (∠ 1 ≅ ∠ 3, ∠ 2 ≅ ∠ 4), we have that ΔABC ≅ ΔCDE by the ASA (angle-side-angle) theorem.

The congruence of those triangles means that segments AB and CD are congruent, thus B bisects AF, and BC &#8773 DE, thus D bisects EF.

So the coordinates of C are x = the midpoint of AF (the average of the x-coordinates of A and E) and y = midpoint of ED (the average of the y-coordinates of A and B. If two angles and the connecting side of one triangle are congruent to the corresponding two angles and connecting side of another, then the two triangles are identical ### Practice problems

Calculate the midpoint of the segments with the following endpoints:

December 4, 2020 by sastry

## Midpoint of a Line Segment

The point halfway between the endpoints of a line segment is called the midpoint. A midpoint divides a line segment into two equal segments. By definition, a midpoint of a line segment is the point on that line segment that divides the segment two congruent segments.
In Coordinate Geometry, there are several ways to determine the midpoint of a line segment.

Method 1:
If the line segments are vertical or horizontal, you may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints.

Find the midpoints $$\overline < AB >$$ and $$\overline < CD >$$. AB is 8 (by counting). The midpoint is 4 units from either endpoint. On the graph, this point is (1,4).
CD is 3 (by counting). The midpoint is 1.5 units from either endpoint. On the graph, this point is (2,1.5)

Method 2:
If the line segments are diagonally positioned, more thought must be paid to the solution. When you are finding the coordinates of the midpoint of a segment, you are actually finding the average (mean) of the x-coordinates and the average (mean) of the y-coordinates.

This concept of finding the average of the coordinates can be written as a formula: NOTE: The Midpoint Formula works for all line segments: vertical, horizontal or diagonal. Consider this “tricky” midpoint problem:
M is the midpoint of $$\overline < CD >$$. The coordinates M(-1,1) and C(1,-3) are given. Find the coordinates of point D.
First, visualize the situation. This will give you an idea of approximately where point D will be located. When you find your answer, be sure it matches with your visualization of where the point should be located.  Other Methods of Solution:
Verbalizing the algebraic solution:
Some students like to do these “tricky” problems by just examining the coordinates and asking themselves the following questions:
“My midpoint’s x-coordinate is -1. What is -1 half of? (Answer -2)
What do I add to my endpoint’s x-coordinate of +1 to get -2? (Answer -3)
This answer must be the x-coordinate of the other endpoint.”
These students are simply verbalizing the algebraic solution.
(They use the same process for the y-coordinate.)

Utilizing the concept of slope and congruent triangles:
A line segment is part of a straight line whose slope (rise/run) remains the same no matter where it is measured. Some students like to look at the rise and run values of the x and y coordinates and utilize these values to find the missing endpoint.

Find the slope between points C and M. This slope has a run of 2 units to the left and a rise of 4 units up. By repeating this slope from point M (move 2 units to the left and 4 units up), you will arrive at the other endpoint. By using this slope approach, you are creating two congruent right triangles whose legs are the same lengths. Consequently, their hypotenuses are also the same lengths and DM = MC making M the midpoint of $$\overline < CD >$$.

I was asked to proof that the midpoint formula of a line works. Obviously it works, but how do I go about proofing that. I started off with a line PQ at a slope, and formed a right triangle. Next, the point M is b/n P and Q, and I guess I’m trying to show M is in the middle. After this, I’m kinda lost, so anything to get me back on track will be appreciated.

Thanks in advance, I loved the work that you are doing here 🙂

One approach is to use the distance formula to calculate the relevant distances. The distance between P and Q is The distances between M and P and M and Q are, respectively, and Note that the distance from M to P simplifies to become which is just half the distance from Q to P . It is easy to see that the same holds true for the distance from M to P and therefore M is indeed the midpoint of the segment PQ .

Another approach is to make use of similar triangles. Let M be the midpoint of the line segment PQ (i.e. the point which is exactly half way between the two points). Now draw vertical and horizonal lines through all of the points. We will assume here that PQ is neither vertical nor horizontal, since it is simple to show that the midpoint formula is true for horizonal and vertical line segments.

The horizonal line though P and the vertical line through Q will meet at a unique point R . Similarly the horizontal line through M meets the vertical line through Q at a single point A and the horizontal line through P meets the vertical line through M at a single point B .

Note that PMB and MQA are similar triangles. Since | PM | = | MQ | (because M is the midpoint of PQ ), they are congruent triangles. Thus | PB | = | MA | = | BR | so B is the midpoint of PR , and similarly A is the midpoint of QR .

The x -coordinate of M equals the x -coordinate of B . Assuming R is to the right of P as in the picture, this equals the x -coordinate of P plus | PB |, which is x + | PB | = x + (1/2)| PR | = x + (1/2)( X – x ) = ( x + X )/2. (If R is to the left of P , so is B , and we get the x -coordinate of M equal to x – | PB | = x – (1/2)| PR | = x + (1/2)( x – X ) = ( x + X )/2).

A similar argument shows the y -coordinate of M equals ( y + Y )/2.

## Section 9.2 The Distance and Midpoint Formulas

### Subsection Distance in a Coordinate Plane

Figure (a) shows a line segment joining the two points $$(-2, 7)$$ and $$(6, 3)\text<.>$$ What is the distance between the two points?

The distance between two points is the length of the segment joining them.

If we make a right triangle as shown in Figure (b), we can use the Pythagorean theorem to find its length. First, notice that the coordinates at the right angle are $$(-2, 3)\text<.>$$ We can find the lengths of the two legs of the triangle, because they are horizontal and vertical segments.

The segment we want is the hypotenuse of the right triangle, so we apply the Pythagorean theorem.

### Subsection The Distance Formula

We can also use the Pythagorean theorem to derive a formula for the distance between any two points, $$P_1$$ and $$P_2\text<,>$$ in terms of their coordinates. We first label a right triangle, as we did in the exmple above. Draw a horizontal line through $$P_1$$ and a vertical line through $$P_2\text<.>$$

These lines meet at a point $$P_3\text<,>$$ as shown in the figure below. The $$x$$-coordinate of $$P_3$$ is the same as the $$x$$-coordinate of $$P_2\text<,>$$ and the $$y$$-coordinate of $$P_3$$ is the same as the $$y$$-coordinate of $$P_1\text<.>$$ Thus, the coordinates of $$P_3$$ are $$(x_2, y_1)\text<.>$$

The distance between $$P_1$$ and $$P_3$$ is $$\abs$$ , and the distance between $$P_2$$ and $$P_3$$ is $$\abs \text<.>$$ These two numbers are the lengths of the legs of the right triangle. The length of the hypotenuse is the distance between $$P_1$$ and $$P_2\text<,>$$ which we’ll call $$d\text<.>$$ By the Pythagorean theorem,

Taking the (positive) square root of each side of this equation gives us the .

###### Distance Formula.

The $$d$$ between points $$P_1(x_1,y_1)$$ and $$P_2(x_2,y_2)$$ is

###### Example 9.12 .

Find the distance between $$(2,-1)$$ and $$(4,3)\text<.>$$

Substitute $$(2,-1)$$ for $$(x_1,y_1)$$ and $$(4,3)$$ for $$(x_2,y_2)$$ in the distance formula to obtain

It doesn’t matter which point we call $$P_1$$ and which is $$P_2\text<.>$$ We obtain the same answer in the previous Example if we switch the two points and use $$(4,3)$$ for $$P_1$$ and $$(2,-1)$$ for $$P_2\text<:>$$

###### Caution 9.13 .

We cannot simplify $$\sqrt <4+16>$$ as $$\sqrt<4>+\sqrt <16>\text<.>$$ Remember that $$\sqrt\ne a+b \text<.>$$ You can easily see this by observing that

so it cannot be true that $$\sqrt <3^2+4^2>$$ equals $$3+4\text<,>$$ or 7. For the same reason, we cannot simplify the distance formula to $$(x_2-x_1)+(y_2-y_1)\text<.>$$

###### Checkpoint 9.14 . Practice 1.

Graph for Practice 1:

### Subsection Finding the Midpoint

The of a segment is the point halfway between its endpoints, so that the distance from the midpoint to either endpoint is the same. The $$x$$-coordinate of the midpoint is halfway between the $$x$$-coordinates of the endpoints, and likewise for the $$y$$-coordinate. For the points $$(-2, 7)$$ and $$(6, 3)$$ shown below, the $$x$$-coordinate of the midpoint is $$2\text<,>$$ which is halfway between $$-2$$ and $$6\text<.>$$ The $$y$$-coordinate is halfway between $$7$$ and $$3\text<,>$$ or $$5\text<.>$$ Thus, the midpoint is $$(2, 5)\text<.>$$

### Subsection The Midpoint Formula

If we know the coordinates of two points, we can calculate the coordinates of the midpoint. Each coordinate of the midpoint is the average of the corresponding coordinates of the two points.

###### Midpoint Formula.

The of the line segment joining the points $$P_1(x_1,y_1)$$ and $$P_2(x_2,y_2)$$ is the point $$M(\overline,\overline)\text<,>$$ where

###### Example 9.15 .

Find the midpoint of the line segment joining the points $$(-2,1)$$ and $$(4,3)\text<.>$$

We substitute $$(-2,1)$$ for $$(x_1,y_1)$$ and $$(4,3)$$ for $$(x_2,y_2)$$ in the midpoint formula to obtain

The midpoint of the segment is the point $$(\overline,\overline)=(1,2)\text<.>$$

In these lessons, we will learn

• the midpoint formula.
• how to find the midpoint given two endpoints.
• how to find one endpoint given the midpoint and another endpoint.
• how to proof the midpoint formula.

We have included a midpoint calculator at the end of this lesson.

### The Midpoint Formula

Some coordinate geometry questions may require you to find the midpoint of line segments in the coordinate plane. To find a point that is halfway between two given points, get the average of the x-values and the average of the y-values.

The following diagram shows the midpoints formula for the two points (x1,y1) and (x2,y2). Scroll down the page for more examples and solutions on how to use the midpoint formula. For Example:
The midpoint of the points A(1,4) and B(5,6) is

### Find the midpoint given two endpoints

We can use the midpoint formula to find the midpoint when given two endpoints.

Example:
Find the midpoint of the two points A(1, -3) and B(4, 5).

Solution:
Midpoint = = (2.5, 1)

How to use the formula for finding the midpoint of two points?

Example:
Find the midpoint of the two points (5, 8) and (-5, -6).

How to use the midpoint formula given coordinates in fractions?

Example:
Determine the midpoint of the two points (2/3, 1/4) and (11/6, 7/9).

### Find an endpoint when given a midpoint and another endpoint

We can use the midpoint formula to find an endpoint when given a midpoint and another endpoint.

Example:
M(3, 8) is the midpoint of the line AB. A has the coordinates (-2, 3), Find the coordinates of B.

Solution:
Let the coordinates of B be (x, y)

Coordinates of B = (8, 13)

How to find a missing endpoint when given the midpoint and another endpoint?

How to solve problems using the Midpoint Formula?

Example:
For a line segment DE, one endpoint is D(6, 5) and the midpoint M(4, 2). Find the coordinates of the other endpoint, E.

### Proof of the Midpoint Formula

How to derive the midpoint formula by finding the midpoint of a line segment?

How to use the Pythagorean theorem to prove the midpoint formula?
The following video gives a proof of the midpoint formula using the Pythagorean Theorem.
Step 1: Use the distance formula to show the midpoint creates two congruent segments.
Step 2: Use the slope formula to show that the coordinate of the midpoint is located on the line segment.

Midpoint Calculator
Enter the coordinates of two points and the midpoint calculator will give the midpoint of the two points. Use this to check your answers.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. ## Introduction to Midpoint Calculator

Midpoint is the center value of numers or anything. It is like cutting an eight inch pizza whose midpoint will be 4. Calculation of midpoint is same like as we calculate the average online of two numbers by adding together and dividing by two.

Midpoint calculator is an online tool to calculate midpoint from different numbers. You will find the midpoint calculator useful when there are big numbers and you’ll need to find out the middle value or midpoint.

Related: Learn about formulas, similarities and differences between variance and covariance.

## Midpoint of a Line Segment

In the below image there is the midpoint of a line segment that connects two points by finding the point. Midpoint of a line segment calculator is built in a way it can find endpoint from midpoint and endpoint easily. If you want to find the exact mean value, try mean finder maths.

That point is directly in the middle of the line segment such that it cuts it into two congruent halves. In this case, you have line segment JK and point M is directly in the middle. So, JM is ½ and KM is the other half. They are both congruent.

Midpoint calculator supports to calculate the middle of any two points A and B on a line segment. Actually midpoint formula calculator uses coordinates of two points as like $$A(x^A,y^A)A(x^A,y^A)$$ as well as $$B(x^B,y^B)B(x^B,y^B)$$ “X” horizontally and “Y” in parallel

Now let’s discuss about midpoint calculator briefly how its mechanism works. To learn about the significant values and to calculate these, use sig fig arithmetic calculator.

## Midpoint Formula used in the Mid Point Calculator

Now, we are going to find endpoint from midpoint and endpoint on the coordinate plane. So we want to think of a midpoint as a location with XY (x,y) coordinates and our endpoint calculator to find endpoint from midpoint and endpoint using midpoint formula

(xm, ym) means coordinates of the midpoint

(x1, y1) means coordinates of the first point

(x2, y2) means coordinates of the second point

Midpoint or endpoint calculator deals with finding the mid value. It will not find the distance of a value, as the distance is not required for its working. If you need to find the distance formula, try distance formula calculator with steps to find the exact value.

So let’s go ahead and learn how to use it.

## Example 1

In this example we will know about that how to find midpoint.

AB has endpoints at (7, 3) and (-5,5). Plot point M the midpoint of AB.

In this example, we want to find the midpoint of AB and it’s giving us the coordinates (x, y) of both endpoints. So let’s start by plotting those endpoints A at 7, 3 and B at -5, 5 and then constructing line segment will be AB.

So, we want to find the midpoint of this line segment manually without using mid point calculator. Again we want to find the x,y coordinate, that is directly in the middle of this line segment. Such that it cuts it into two congruent halves pieces.

Here Coordinates of A are (7,3) and B (-5,5) so, now substitute the right values into the midpoint formula. Now end points A and B are just XY coordinates.

Since, (7,3) (-5,5) here in first point 7 is x1 and 3 is y1 while in second point -5 is x2 and 5 is y2.

By putting values in midpoint formula

So by using those endpoints in the midpoint formula we have found the coordinates of the midpoint of the AB at 1, 4

So the midpoint formula calculator works right according the same way.

Related: Find useful probability and statistics online tools like variance table calculator and sample covariance calculator. These tools can be very helpful for your learning and practice.

## What is Calculatored’s Midpoint Calculator?

Calculatored is an online web portal offering tons of online calculators and free converters. Midpoint formula calculator is one of those tools which you can use without any fee or subscription. Our midpoint of a line segment calculator will instantly calculate the X midpoint and the Y midpoint of different coordinates online. You can use this endpoint calculator to save your time and complete your task or assignments quickly.

## How to use Midpoint Calculator?

Our midpoint and endpoint calculator is simple and easy to use. You will find the midpoint calculator extremely useful in order to find endpoint from midpoint and endpoint. Just fill the values into 4 input fields to get the answer quickly. Midpoint formula calculator also provides help to solve these types of advance problems. Note, Midpoint formula calculator and mid point calculator are different names of the same mechanism.

If anyone want to find endpoint from midpoint, endpoint calculator is the best online tool available for this purpose.

Also our online tools isn’t an unknown endpoint calculator or any other calculation tool, our tool is accurate and reliable and you can find midpoint calculator online on Google.

The midpoint rule calculator practices the midpoint each interval as the point at which estimate the function for the Rieman sum. In actual Riemann sum, the values of the function and height of each rectangle is equal at the right endpoint while in a midpoint Riemann sum, rectangle height is equal to the value of the function at its midpoint.

To find arithmetic sequence, use common difference arithmetic sequence calculator and to find sphere volume, try out volume sphere calculator.

We hope this article will be helpful regarding to understand of the working of midpoint calculator. You can also find the complete tutorial of midpoint of line segment calculator for your advance understanding. Please provide your valuable feedback. Cheers!

A line segment is a part of a line that has two endpoints and a fixed length. It is different from a line that does not have a beginning or an end and which can be extended in both directions. In this lesson, we will learn more about a line segment, its symbol, and the way to measure a line segment.

 1 What is a Line Segment? 2 How to Measure Line Segments? 3 Line Segment Formula 4 Difference Between Line, Line Segment, and Ray 5 FAQs on Line Segment

## What is a Line Segment?

A line segment is a path between two points that can be measured. Since line segments have a defined length, they can form the sides of any polygon. The figure given below shows a line segment AB, where the length of line segment AB refers to the distance between its endpoints, A and B.

### Line Segment Symbol

A line segment is represented by a bar on top which is the line segment symbol. It is written as $$\overline$$. ## How to Measure Line Segments?

Line segments can be measured with the help of a ruler (scale). Let us see how to measure a given line segment and name it PQ.

• Step 1: Place the tip of the ruler carefully so that zero is placed at the starting point P of the given line segment.
• Step 2: Now, start reading the values given on the ruler and spot the number which comes on the other endpoint Q.
• Step 3: Thus, the length of the line segment is 4 inches, which can be written as $$\overline$$ = 4 inches. ## Line Segment Formula

In the above example, we measured the length of line segment PQ to be 4 inches. This is written as $$\overline$$ = 4 inches. Now, let us see how to find the length of a line segment when the coordinates of the two endpoints are given. In this case, we use the distance formula, that is, D = √[($$x_<2>-x_<1>$$) 2 + ($$y_<2>-y_<1>$$) 2 ]. Here, ($$x_<1>$$, $$y_<1>$$) and ($$x_<2>$$, $$y_<2>$$) are the coordinates of the given points.

For example, a line segment has the following coordinates: (-2, 1) and (4, –3). Let us apply the distance formula to find the length of the line segment. Here, $$x_<1>$$ = -2; $$x_<2>$$ = 4; $$y_<1>$$ = 1; $$y_<2>$$ = -3. After substituting these values in the distance formula we get: D =√[(4-(-2)) 2 + (-3-1) 2 ) = √((4+2) 2 + (-3-1) 2 ] = √(6 2 + (-4) 2 ) = √(36 + 16) = √52 = 7.21 units. Therefore, using the distance formula, we found that the length of the line segment with coordinates (-2, 1) and (4, –3) is 7.21 units.

## Difference Between Line, Line Segment, and Ray

Observe the figures given below to understand the difference between a line, a line segment, and a ray. A line is a set of points that extends in two opposite directions indefinitely.

A line segment is a part of a line having a beginning point and an endpoint.

A ray is a part of a line that has a start point but no definite endpoint.

It has no endpoints and is written as $$\overleftrightarrow$$.

Important Notes

• A line has indefinite ends and cannot be measured.
• A line segment has a start point and an endpoint, thus, it can be measured.
• Line segments have a defined length, hence, they form the sides of any polygon.
• A ray has just one start point and no endpoint, therefore, it cannot be measured.
• The concept of rays can be understood with the example of the rays of the sun, which have a beginning point but no endpoint.

☛Topics Related

Check out the following pages related to the line segment.

## Line Segment Examples

Example 1: Identify if the given figure is a line segment, a line, or a ray. Solution:

The figure has one starting point but an arrow on the other end. This shows that it is not a line segment or a line, it is a ray. Therefore, LM is a ray.

Example 2: Name the line segments in the given triangle. Solution:

The line segments which make up the triangle are $$\overline$$, $$\overline$$, and $$\overline$$.
Therefore, the line segments in the given triangle are $$\overline$$, $$\overline$$, and $$\overline$$.

Example 3: Find the length of the line segment PQ if the coordinates of P and Q are (3, 4) and (2, 0) respectively.

Solution:

The coordinates of P and Q are (3, 4) and (2, 0). Let us apply the distance formula: D = √[($$x_<2>-x_<1>$$) 2 + ($$y_<2>-y_<1>$$) 2 ]. Here, $$x_<1>$$ = 3; $$x_<2>$$ = 2; $$y_<1>$$ = 4; $$y_<2>$$ = 0. Therefore, the length of the line segment, D =√[(2-3) 2 +(0-4) 2 ] = √((-1) 2 +(-4) 2 ) = √(1 + 16) = √17 = 4.123 units.
Therefore $$\overline$$ = 4.123 units.

##### New member
• Apr 13, 2013
• #1
• The coordinates of point T are given. The midpoint of ST is (5, 28). Find the
coordinates of point S.

So how do you find the endpoint when the midpoint is given?

##### Elite Member
• Apr 13, 2013
• #2
• \left(s_x,s_y\right)\) is $$\displaystyle \left(\dfrac<2>,\dfrac<2>\right)$$.

In other words, just average the coordinates.

#### lookagain

##### Elite Member
• Apr 13, 2013
• #3
• \left(s_x,s_y\right)\) is $$\displaystyle \left(\dfrac<2>,\dfrac<2>\right)$$.

In other words, just average the coordinates.

##### Elite Member
• Apr 13, 2013
• #4
• #### wjm11

##### Senior Member
• Apr 13, 2013
• #5
• The coordinates of point T are given. The midpoint of ST is (5, 28). Find the
coordinates of point S.

So how do you find the endpoint when the midpoint is given?

PKA has provided information to approach the problem algebraically. Let’s try another approach to understanding this problem.

Using graph paper, make an xy graph and plot the two points T(0,4) and M(5,28). [If you do not have graph paper, you can still make a sketch of this graph and points to help you understand.]

Place your pencil on point T. Move your pencil to the right 5 units. The pencil is now directly under the midpoint M. Now move your pencil straight up 24 units. You are now on the midpoint, M.

If you repeat these moves starting at point M (going 5 to the right, then 24 up) you will be at the other end point, S.

Does this make sense to you?

Ask yourself: How did we know how far to move each time? Well, look at the x and y values of points T and M. For the x direction, we have 5 – 0 = 5. For the y direction, we have 28 – 4 = 24. Thus, if we add those values to the point M values, we end up at the other end point: 5 + 5 = 10 (for x) and 28 + 24 = 52 (for y).

Midpoint formula worksheets have a wide range of high school practice pdfs to find the midpoint of a line segment using number lines, grids and midpoint formula method. Also determine the missing coordinates, midpoint of the sides or diagonals of the given geometrical shapes, missing endpoints and more. Free pdf worksheets are also included. In these printable midpoint worksheets a number line is marked with numerous points. Find the midpoint of each indicated line segment. Gain a basic idea of finding the midpoint of a line segment depicted on a grid with this exercise! Locate the endpoints of the line segments, add both x-coordinates and divide by 2, and repeat the steps with y-coordinates as well to obtain the coordinates of the midpoint. Level up finding the midpoint of a line segment whose endpoints are located on different quadrants of a coordinate grid. Analyze the x and y-axes, find the locations of the endpoints, calculate the position of the midpoint, and write it as an ordered pair. Define the formula for the midpoint of two endpoints (x1, y1) and (x2, y2), as M = [(x1 + x2)/2, (y1 + y2)/2], and direct high school students to apply it and solve the problems here. Gain momentum along your way in using the midpoint formula to determine the point halfway between indicated points with integer, fractional, and decimal values. Find the center of a circle, median of a triangle, point of intersection of diagonals of a square, rectangle, parallelogram and more in these printable high school worksheets. The endpoints and the midpoint of a line segment are given with an unknown value. Find the value of the unknown. In these pdf worksheets the one end of a line segment and the midpoint are given. Find the other end of the line segment Challenge students to figure out the ordered pairs of a point that is at a fractional distance from another indicated point. Also, ask them to find the missing coordinates of points on geometric figures.

The midpoint of a line segment is the point that divides the segment into two congruent segments. Congruent segments are segments that have the same length.

You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints.

## Midpoint Formula The midpoint M of the line segment AB is

## Finding the Coordinates of a Midpoint

Find the coordinates of the midpoint of the line segment CD with endpoints C(-2, -1) and D (4, 2). Write the formula.

## Finding the Coordinates of an Endpoint

M is the midpoint of the line segment AB. A has coordinates (2, 2), and M has coordinates (4, -3). Find the coordinates of B.

Let the coordinates of B equal (x, y).

Use the Midpoint Formula.

(4, -3) = [(2 + x)/2, (2 + y)/2]

Find the x-coordinate.

Find the y-coordinate.

The coordinates of B are (6, вЂ“8).

Graph points A and B and midpoint M. Point M appears to be the midpoint of the line segment AB.

You can also use coordinates to find the distance between two points or the length of a line segment.

To find the length of segment PQ, draw a horizontal segment from P and a vertical segment from Q to form a right triangle as shown below. Solve for c. Use the positive square root to represent distance.

This equation represents the Distance Formula.

## Distance Formula

In a coordinate plane, formula to find the distance between two points (x 1 , y 1 ) and (x 2 , y 2 ) is

## Finding Distance in the Coordinate Plane

Use the Distance Formula to find the distance, to the nearest hundredth, from A(-2, 3) to B(2, -2). Substitute (-2, 3) for (x 1 , y 1 ) and (2, -2) for (x 2 , y 2 ).

d = в€љ[(2 + 2) 2 + (-2 – 3) 2 ]

The distance between from A(-2, 3) to B(2, -2) is about 6.40 units.

## Geography Application

Each unit on the map of Lake Okeechobee represents 1 mile. Kemka and her father plan to travel from point A near the town of Okeechobee to point B at Pahokee. To the nearest tenth of a mile, how far do Kemka and her father plan to travel? Substitute (33, 13) for (x 1 , y 1 ) and (22, 39) for (x 2 , y 2 ).

d = в€љ[(33 – 22) 2 + (13 – 39) 2 ]

d = в€љ[11 2 + (-26) 2 ]

Kemka and her father plan to travel about 28.2 miles.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here. ###### Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

• Explanation
• Transcript

There are two formulas that are important to remember when considering vectors or positions in the 3D coordinate System. The midpoint formula and the distance formula in 3D. The midpoint and distance formula in 3D can be derived using a method of addition of the geometric representation of vectors. In order to understand the derivation of the distance formula in 3D we must understand 3D vector operations.

I want to derive the midpoint formula for 3 dimensions, the midpoint formula is going to help me find the midpoint between points a which is coordinates x1, y1 and z1 and b which has coordinates x2, y2 and z2. So I have segment ab drawn here and I’ve labeled my midpoint m and I’m hoping to find the formula for it’s coordinates. I’ve also added a position vector oa for point a and a position vector om for point m. Now let’s find the components for position vector oa, and let’s recall that the components of a position vector are exactly the coordinates of the endpoint of that vector so there are going to be x1, y1, z1 and I’m also going to need vector ab in order to find m and what are the components of ab? Well since vector ab goes from point a to point b and the components are x2-x1, y2-y1 and z2-z1 okay how are we going to get position vector om from oa and ab?
Well let’s make the observation that the vector that starts at point a and ends at m is half of the vector that goes from a to be, so this is vector ab this vector starting here and ending here is a half of ab the scalar of multiple ohe half of ab and so I need to add that to oa to get om. So vector om=oa plus one half a b, so that’s going to be in components x1, y1, z1 plus one half of this, one half of x2-x1, y2-y1 and z2-z1. So let’s see if we can combine this in a single step for the first component I’m going to get x1 plus a half x2 minus a half x1 so a half x1 plus a half x2 now I’ll get y1 plus a half y2 minus a half y1 that’s one half y1 plus one half y2 and similarly I get one half z1 plus one half z2.
Each of these is exactly the average of the x and y components of these 2 points, so I can write it as x1+x2 over 2 y1+y2 over 2 and z1+z2 over 2, these are the components of vector om which goes from the origin to point m and therefore the coordinates of point m are these. So the midpoint m of the segment joining x1, y1, z1 and x2, y2, z2 are x1+x2 over 2 y1+y2 over 2 and z1+z2 over 2 and that’s the midpoint formula.

Segments and rays are under the subset of lines. A segment is a part of a line having two endpoints and has a particular length. On the other hand, a ray is also a part of line having one endpoint and the other direction extends indefinitely. Figure 1 illustrates a segment and a ray. ### Terms to remember

• Congruent

– having the same size and shape.

– a numerical value that describes how far two objects are.

– a point located at the end part of a segment, defining its limit.

– reverse in position or direction.

– a portion of a set.

### Definition of a Segment

A segment is a set of points consisting of two points of the line called the endpoints, and all of the points of the line between the endpoints. It is commonly used to represent the length, height, or width of a certain object and the distance between two objects. It is named by using the label of its endpoints and insert a line (( ) ̅) above the letters. Figure 2 shows segment AB which can also be written as (AB) ̅. ##### Example 1

Draw (CD) .

Draw two points and label it as C and D.

Connect the two points straight each other.

##### Example 2

How many segments are there in the line below?

Use the points in the line as endpoints of the segments.

The segments are (QR) , (RS) , (ST) , (QS) , (RT) , and (QT) .

### Midpoint of a Segment

A midpoint of a segment is the point that divides the line segment into two congruent parts. It is located at the center of the segment. Figure 3 shows the midpoint of a segment.

Note: The two vertical lines indicate that the distances from the midpoint to both endpoints are equal.

##### Example 3

The length of (XY) is 12 and Z is the midpoint of (XY) . Find the length of (XZ) .

The midpoint Z divides (XY) into two congruent parts: (XZ) and (YZ) .

The length of (XZ) is ½ the length of (XY) which is ½ 12 = 6.

##### Example 4

O is the midpoint of (NP) . If (NO) = 9, what is the length of (NP) ?

The midpoint O divides (NP) ̅ into two congruent parts: (NO) ̅ and (PO) ̅.

sThe length of (NP) ̅ is twice the length of (NO) ̅ which is 2 9 = 18.

### Addition and Subtraction of Segments

Figure 4 illustrates three collinear points E, F, and G forming a segment. Points E and G constitute the endpoints of the segment which is (EG) and point F in between divides (EG) into two segments: (EF) and (FG) . The sum of the lengths of (EF) and (FG) is equal to the length of (EG) . Therefore, (EF) + (FG) = (EG) . The expression represents a segment if point F is between Points E and G.

The following expressions are also true for the lengths of the segments:

(EF) = (EG) – (FG)

(FG) = (EG) – (EF)

##### Example 5

Find the length of (JL) in the figure.

Points J and L are endpoints of (JL) and point K is between points J and L.

Therefore, (JK) + (KL) = (JL) .

##### Example 6

The length of (UV) is 13 and W is between points U and V. If (WU) = 5, what is the length of (VW) ?

Points U and V are endpoints of (UV) and point W is between points U and V.

Therefore, (UW) + (VW) = (UV) .

So, (VW) = (UV) – (UW) = 13 – 5 = 8.

### Definition of a Ray

A ray consists of a point on a line and all points on one side of the point. It has only one endpoint. Rays are commonly used in physics to denote direction and also force. In naming a ray, consider two points in the ray: one is the endpoint and the other is any point in the ray. The label of the endpoint should be the first letter of the name of the ray and place a right arrow sign ( ( ) ) above the letters. Figure 5 illustrates a ray labeled as (RY) . Opposite rays are two rays that lie on the same line having a common endpoint and no other point in common. In Figure 6 shown, (RS) ̅ and (ST) ̅ are opposite rays.

##### Example 7

Draw (AE) .

Draw two points and labeled it as A and E.

Connect the two points straight each other and extend the line from point E.

##### Example 8

Find the number of rays that can be found in the line.

Explanation: 12

Consider two points in the line as the points of the ray.

Rays pointing to the right can be named as (MB) , (BP) , (PC) , (MP) , (BC) , and (MC) .

Rays pointing to the left can be named as (CP) ,(PB) , (BM) , (CB) , (PM) , and (CM) .

There are 12 rays that are found in the line.

The midpoint of a segment is the point on the segment that is equidistant from the endpoints.
The midpoint is the point on the segment halfway between the endpoints .

It may be the case that the midpoint of a segment can be found simply by counting.

Example 1: Given the graph at the right, find the midpoint of and of .
AB
= 4 units (by counting). The midpoint is 2 units from either endpoint. This midpoint is (-1,2).

CD = 5 units (by counting). The midpoint is 2.5 units from either endpoint. This midpoint is (3,-1.5). The midpoint formula works for ALL line segments: vertical, horizontal or diagonal.

The midpoint formula finds an ordered pair.

Example 2: Find the midpoint of segment , where A(-3,-3) and B(1,4).

Using the midpoint formula, we have

Note: Fractional answers should be left as fractions or written as decimals if terminating. Do not round these midpoint coordinates, unless told to do so. By examining the graph, we can see approximately where point D will be located. When you solve for the coordinates of D, be sure they match with your approximation of point D. In questions such as Example 3, you may be able to find the missing endpoint by using slope and congruent triangles.

Example 4: Given M is the midpoint of , as shown, find the coordinates of point B.

If the midpoint and the endpoint coordinates are integer values (making them easy to locate), you can build congruent triangles to find the missing endpoint. We know that is a straight line segment whose slope (rate of change) will be constant along the segment.
The slope between points A and M runs 4 units to the right and 2 units up. By repeating this slope from point M (4 units right and two units up), we will find the missing endpoint B (4,0). This approach is harder to implement when the coordinates are fractional. If you are dealing with fractional coordinates, use the midpoint formula.