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## What Does Segment Length Mean?

The distance between two points on a line segment is the segment length.

It is a very concise definition. For short, segment length is all about "from one point to another". Recall the mountaineering segments, those were just parts of the total distance that we had to cover.

Meanwhile, one cannot understand segment length without taking into perspective points, because these are your focus on locating where the segment starts as well as where it stops.

## What is the length of a segment between two points?

The length of a segment between two points is the distance between two points. One point serves as the starting point which is the location from which the measurement begins. Meanwhile, the other is the endpoint which is the location where the measurement stops. It is sometimes a name with a small letter or to letters in upper cases. For instance, if there are two points A and B, we can call the length segment existing between A and B, or c, or we may just call the line segment, AB.

### What are the coordinates that exist on the length of a segment between two points?

Since we are dealing with points, we need to know their position on the cartesian plane. In other words, we should know the position of the starting and ending point on the x and y-axis. This position is referred to as the coordinates of the points of the length segment and they are written in the form (x_{1}, y_{1}) and (x_{2}, y_{2}).

Here,

x_{1} means the position of the starting point in the x-axis,

y_{1} means the position of the starting point in the y-axis,

x_{2} means the position of the ending point in the x-axis

and

y_{2} means the position of the ending point in the y-axis.

The image below describes this clearly.

Now we can see the length segment as not only a distance, but we now consider the points that determine this distance.

You should now think of how to determine the length of segments when you know their starting and ending points.

## What is the Formula for Segment Length Between Two Coordinates?

To find the segment length, we can create a right-angled triangle and hence use Pythagoras' theorem to solve for the distance:

We can see that $\u2206y$ is the vertical distance between points A and B. $\u2206x$ is the horizontal distance between points A and B. Hence, we can complete a Pythagorean triangle by inserting the distance between points A and B as $d$.

Using Pythagoras' theorem, we know:

${d}^{2}=\u2206{y}^{2}+\u2206{x}^{2}$

Since the distance between two points cannot be negative, we know:

$d=+\sqrt{\u2206{y}^{2}+\u2206{x}^{2}}$

For points A=(${x}_{1},{y}_{1}$) and B=(${x}_{2},{y}_{2}$):

$\u2206y={y}_{2}-{y}_{1}$ and $\u2206x={x}_{2}-{x}_{1}$

Therefore:

$d=+\sqrt{{({y}_{2}-{y}_{1})}^{2}+{({x}_{2}-{x}_{1})}^{2}}$

Note that since ∆y and ∆x are both squared, there is no need to take the absolute value of these numbers as squaring them turns them positive. Also, note that the square root does not cancel the fact that ∆y and ∆x are squared as the equation adds these terms and does not multiply them.

Find the distance between the points A=(5, 0) and B=(3,7)

**Solution:**

Substituting the coordinates into the equation for segment length:

$d=+\sqrt{{(7-0)}^{2}+{(3-5)}^{2}}$

$d=+\sqrt{{7}^{2}+{(-2)}^{2}}$

$d=+\sqrt{49+4}$

$d=+\sqrt{53}$

$d=7.28(2d.p.)$

Unless specified, you may choose to leave your answer in exact or numerical form.

## Length of a segment with endpoints

In some cases, you may be given only the endpoints and midpoints and you would need to determine the length of the whole segment.

The midpoint is the point halfway through the distance between the starting and the ending point.

When this occurs, the first step to follow is to find the starting point which was not given initially. So, for a starting point A(x_{1}, y_{2}), midpoint M (x_{m}, y_{m}) and endpoint B (x_{2}, y_{2}), the midpoint for x-axis is calculated as:

${\mathit{x}}_{\mathbf{m}}\mathbf{=}\frac{{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{x}}_{\mathbf{2}}}{\mathbf{2}}$

and the midpoint for the y-axis is calculated as:

${\mathit{y}}_{\mathbf{m}}\mathbf{=}\frac{{\mathbf{y}}_{\mathbf{1}}\mathbf{+}{\mathbf{y}}_{\mathbf{2}}}{\mathbf{2}}$

However, our interest is in finding the starting point when only the endpoint and the midpoint are given. In this case, you just need to make in their respective cases x_{1} or y_{1} the subject of the formula. This means that the coordinate of the starting point in the x-axis, x_{1} is:

${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{2}{\mathit{x}}_{\mathbf{m}}\mathbf{-}{\mathit{x}}_{\mathbf{2}}$

Solved as

${x}_{m}=\frac{{x}_{1}+{x}_{2}}{2}\phantom{\rule{0ex}{0ex}}\overline{){x}_{m}=\frac{{x}_{1}+{x}_{2}}{2}}\phantom{\rule{0ex}{0ex}}2{x}_{m}={x}_{1}+{x}_{2}\phantom{\rule{0ex}{0ex}}{x}_{1}=2{x}_{m}-{x}_{2}$

and the coordinate of the starting point in the y-axis, y_{1} is:

${\mathit{y}}_{\mathbf{1}}\mathbf{=}\mathbf{2}{\mathit{y}}_{\mathbf{m}}\mathbf{-}{\mathit{y}}_{\mathbf{2}}$

Solved as

${y}_{m}=\frac{{y}_{1}+{y}_{2}}{2}\phantom{\rule{0ex}{0ex}}\overline{){y}_{m}=\frac{{y}_{1}+{y}_{2}}{2}}\phantom{\rule{0ex}{0ex}}2{y}_{m}={y}_{1}+{y}_{2}\phantom{\rule{0ex}{0ex}}{y}_{1}=2{y}_{m}-{y}_{2}$

If Nonso is on a journey in which his path is linear and he has currently covered half of the distance. If his current coordinate is (4, -2) and his journey terminates at K (9, 5), find the segment length of the whole journey.

**Solution:**

From the information given, Nonso is currently at the midpoint of the total journey which is the segment length of the journey. Since K is where the journey ends that means we have our endpoint. With these, we can now find the coordinates of our starting point as

${x}_{1}=2{x}_{\mathrm{m}}-{x}_{2}\phantom{\rule{0ex}{0ex}}{x}_{\mathrm{m}}=4\phantom{\rule{0ex}{0ex}}{x}_{2}=9\phantom{\rule{0ex}{0ex}}{x}_{1}=2\left(4\right)-9\phantom{\rule{0ex}{0ex}}{x}_{1}=8-9\phantom{\rule{0ex}{0ex}}{x}_{1}=-1$

and

${y}_{1}=2{y}_{\mathrm{m}}-{y}_{2}\phantom{\rule{0ex}{0ex}}{y}_{\mathrm{m}}=-2\phantom{\rule{0ex}{0ex}}{y}_{2}=5\phantom{\rule{0ex}{0ex}}{y}_{1}=2(-2)-5\phantom{\rule{0ex}{0ex}}{y}_{1}=-4-5\phantom{\rule{0ex}{0ex}}{y}_{1}=-9$

This means that Nonso started his journey at the point (-1,-9).

Now we know his starting point, we can calculate the length of the segment for the journey as:

$d=+\sqrt{{({y}_{2}-{y}_{1})}^{2}+{({x}_{2}-{x}_{1})}^{2}}\phantom{\rule{0ex}{0ex}}d=+\sqrt{{(5-(-9\left)\right)}^{2}+{(9-(-1\left)\right)}^{2}}\phantom{\rule{0ex}{0ex}}d=+\sqrt{{(5+9)}^{2}+{(9+1)}^{2}}\phantom{\rule{0ex}{0ex}}d=+\sqrt{{14}^{2}+{10}^{2}}\phantom{\rule{0ex}{0ex}}d=+\sqrt{196+100}\phantom{\rule{0ex}{0ex}}d=\sqrt{296}\phantom{\rule{0ex}{0ex}}\mathit{d}\mathbf{=}\mathbf{17}\mathbf{.}\mathbf{2}$

## What is the length of a segment of a circle?

A segment of a circle is bounded by an arc and a chord. The line segment of a circle can either be the diameter of a circle when the line passes through the center of the circle or a chord if the line passes any other place apart from the center of a circle.

To calculate the length of the segment of a circle when it passes the center, multiply the given radius by 2. However, when it passes outside the center then the length of a segment of a circle is the length of the chord calculated as

$d=2r\times \mathrm{sin}\left(\frac{\theta}{2}\right)$

Where r is the radius and θ is the angle subtended by the sector that forms the segment.

This formula was gotten from the image description below;

from the image using SOHCAHTOA we get

$\mathrm{sin}\left(\frac{\theta}{2}\right)=\frac{\frac{d}{2}}{r}\phantom{\rule{0ex}{0ex}}r\times \mathrm{sin}\left(\frac{\theta}{2}\right)=\frac{d}{2}\phantom{\rule{0ex}{0ex}}d=2r\times \mathrm{sin}\left(\frac{\theta}{2}\right)$

Find the length of the line segment of a circle with a radius of 10 cm which subtends 120° at the center.

**Solution:**

The length of the line segment is

$d=2r\times \mathrm{sin}\left(\frac{\theta}{2}\right)\phantom{\rule{0ex}{0ex}}d=2\times 10cm\times \mathrm{sin}\left(\frac{120\xb0}{2}\right)\phantom{\rule{0ex}{0ex}}d=2\times 10cm\times \mathrm{sin}(60\xb0)\phantom{\rule{0ex}{0ex}}d=20cm\times 0.866\phantom{\rule{0ex}{0ex}}d=17.32cm\phantom{\rule{0ex}{0ex}}$

## Segment Length - Key takeaways

- A line segment is the distance between two coordinates.
- It is calculated using Pythagoras' theorem.
- The segment length can be calculated when the endpoint and midpoint are given.
- The line segment of a circle is either the diameter or chord depending on if the line passes through the center of a circle.

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##### Frequently Asked Questions about Segment Length

What is a segment length?

It is the distance between two points on a straight line.

Where is the equation for segment length derived from?

It originates from Pythagoras' theorem.

What is the a circle segment?

It is the area within a circle bound by a chord.

How to find length of segments?

Length of segments can be found by simply applying the distance formula when the endpoints are known.

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