# Endpoint Calculator

Welcome to Omni's **endpoint calculator**, where we'll learn **how to find the endpoint of a line segment** if we know its other end and its midpoint. As you might have guessed, this topic is connected to computing the midpoint, and the endpoint formula is **quite similar to the calculations we made there**. But, before we go into details, we'll slowly go through **the endpoint definition** in geometry to better understand what we're dealing with here.

So, sit back, brew yourself a cup of tea for the journey, and **let's get to it**!

## Endpoint definition in geometry

Colloquially speaking, **an endpoint is a point that lies at the end**. We're sure that this statement was as much of a shock to you as it was to us when we heard it first. But, on the other hand, an eggplant doesn't taste like eggs at all, so **you can never be too sure when guessing the meaning of a word**, can you?

However, there are times, like when you're splitting a pizza for several people, where you have to be **a bit more precise**, and who else could we turn to for that if not **mathematicians**?

In its simplest form, the endpoint definition in geometry **focuses on line segments**, i.e., straight lines connecting two points. Yup, you guessed it - **these points are called endpoints**. Note, that according to this definition, **each segment has two endpoints** (unless it's the degenerate case where they are the same point, i.e., the interval is a single point).

For simplicity and calculations, **we'll call one of them the starting point** (as is done in the endpoint calculator). Keep in mind, however, that **the beginning can just as well be the end if you look at it from the other side**.

Now, that sounded **creepily philosophical**, don't you think? But let's leave the "*Who are we and where do we go?* questions for when we can't fall asleep. We should focus on the segments we've mentioned and how to find the endpoints.

## How to find the endpoint?

In order to get the endpoint, **we need to have some point of reference** to begin with. In other words, since we're dealing with a line segment and one of its components, **we need to know what the rest of it looks like**.

The simplest and most common situation is where **we're missing the endpoint while we know the starting point and the midpoint**. The latter is simply, as the name suggests, the point marking the middle of the segment. This is all that we need to find the endpoint; after all, **it must lie at the other end of the midpoint from the starting point**, and be the same distance away.

Therefore, intuitively, we can already **describe geometrically how to find the endpoint**.

- Given the starting point,
`A`

, and the midpoint,`B`

,**draw the line segment**that connects the two. **Draw a line**going farther from`B`

away from`A`

to God-knows-where.**Measure the distance**from`A`

to`B`

and**mark the same distance**from`B`

going the other way.**The point you marked is the endpoint you seek.**- Proceed to do
**a victory dance**.

However, there are people (and we're not suggesting that we are those people) that **don't really enjoy drawing lines that much**. After all, you need a ruler for that, and Lorde is hard to come by... (Yes, that was a terrible joke, and we bow our heads in shame. But with a slight snigger, nonetheless.)

Anyway, **for people who prefer numbers and calculations** (and we might actually be suggesting that we are those people), we'll focus on **how to find the endpoint algebraically** in the next section. Please, don't be afraid of the word 'algebraically' - in a second you'll see how it translates to "*easily and effortlessly*" - **the very motto of our missing endpoint calculator**.

## Endpoint formula

In coordinate geometry, we handle objects that are embedded in what we call **Euclidean space**. It's not too important right now to understand its mathematical definition, but, for our purposes, it's enough to know that this means that **in such spaces, points**, say, `A`

or `B`

, **have two coordinates**: `A = (x₁, y₁)`

and `B = (x₂, y₂)`

.

The numbers `x₁`

and `x₂`

mark the position of the points with respect to the horizontal axis (usually denoted with `x`

's), while `y₁`

and `y₂`

are used for the vertical axis (most often denoted with `y`

's). Together, such a pair of numbers `(x₁, y₁)`

**defines a point in the space**. What is more, the coordinates **help us analyze more complicated objects in our Euclidean space**. For instance, they appear in **the endpoint formula**.

Say that you have a line segment going from `A = (x₁, y₁)`

to... well, we don't yet know. We will now explain **how to find the endpoint** `B = (x₂, y₂)`

**if we know the midpoint** `M = (x, y)`

.

From the definition of a midpoint, we know that **the distance from** `A`

**to** `M`

**must be the same as that from** `M`

**to** `B`

. It's just that `B`

is on the other side. This means that, to find `B`

, it is enough to "*move*" `M`

along the line going through `A`

and `M`

by the same length as that of the segment `AM`

. Or, if you'd like to sound fancy, by the vector `AM`

.

In other words, we have

`x₂ = x + (x - x₁) = 2x - x₁`

, and

`y₂ = y + (y - y₁) = 2y - y₁`

.

To sum it all up, if you like having **all the information you need in one paragraph**, then there it is.

💡 The endpoint of a line segment going from `A = (x₁, y₁)` to a midpoint at `M = (x, y)` is the point `B = (2x - x₁, 2y - y₁)` . |

Note that above we've mentioned the line going through `A`

and `M`

. Such lines are **quite helpful when you're learning how to find the endpoint or the midpoint**. After all, the segment `AB`

is contained on that line. If your exercise or problem requires more information about them, be sure to check out Omni's coordinate geometry calculators and **find the one that suits your needs**!

Phew, that was a long time spent on theory and formulas, don't you think? How about we leave this technical mumbo-jumbo and **see a numerical example**?

## Example: using the endpoint calculator

Say that four months ago, you began posting videos on YouTube. Nothing fancy, just a few cooking recipes that are traditional to your region. It started as a hobby, but **people seem to be enjoying the show** and you see **the number of viewers increasing linearly with time**. Why don't we try to find the missing endpoint with our calculator to check **how many there should be in four month's time**?

First of all, note that although the problem doesn't seem geometrical at all, **we can indeed find the answer using the endpoint definition from geometry**. After all, the starting point, i.e., month zero, was when you began posting the videos, so we were at `0`

viewers at that point. Now, we're at month four, which will be our midpoint (since we want to find the number of viewers in another four months). In other words, **the endpoint will be our answer**.

Say that **currently, you have** `54,000`

**subscribers**, and let's try to translate all this data in such a way that the endpoint calculator will understand what we want from it.

According to the above section, in order to find the answer, **we need the starting point and the midpoint**. Let's denote them by `A = (x₁, y₁)`

and `M = (x, y)`

, respectively. For us, the `x`

's will denote **the number of months in we are**, and `y`

's will be **the number of viewers**. Since our starting point was month zero, and we're currently 4 months in, we have (and can input into the endpoint calculator)

`x₁ = 0`

,

`x = 4`

.

Now it's time for **the subscribers**. Again, the starting point was when we had no one, while right now, after the four months, we're at `54,000`

. Therefore, we have

`y₁ = 0`

,

`y = 54,000`

.

Once we input all this data into the endpoint calculator, **it will spit out the answer**. But let's not reveal it just yet! How about we see **how to find the endpoint ourselves using the endpoint formula**?

Let's grab a piece of paper and recall the information that we've already mentioned above. Our **starting point was at month zero with zero subscribers**, which means that our starting point is `A = (0, 0)`

. Now **we're at month four with** `54,000`

**subscribers**, which is halfway from what we'd like to calculate. This means that our midpoint is `(4, 54,000)`

.

All we need to do now is **use the endpoint formula** from the above section. If we denote the endpoint's coordinates by `B = (x₂, y₂)`

, then

`x₂ = 2*4 - 0 = 8`

,

`y₂ = 2*54,000 - 0 = 108,000`

.

This means that if the trend continues, **we should arrive at** `108,000`

**subscribers in four months**. Now, that's quite a number, if you ask us! Fortunately, it's all done online, so there should be no problem with social distancing. Now go and pursue your cooking dreams!