Segment Length

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.

An image of a segment length, StudySmarter Originals

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 (x₁, y₁) and (x₂, y₂).

Here,

x₁ means the position of the starting point in the x-axis,

y₁ means the position of the starting point in the y-axis,

x₂ means the position of the ending point in the x-axis

and

y₂ means the position of the ending point in the y-axis.

The image below describes this clearly.

A graphical image of two points of a segment length with its coordinates, StudySmarter Originals

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

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:

An image shows the use of Pythagoras Theorem to calculate the length of a segment from two points, StudySmarter Originals

We can see that $∆y$ is the vertical distance between points A and B. $∆x$ 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:

$d^2=∆y^2+∆x^2$

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

$d=+\sqrt{∆y^2+∆x^2}$

For points A=($x_1,y_1$) and B=($x_2,y_2$):

$∆y=y_2-y_1$ and $∆x=x_2-x_1$

Therefore:

$d=+\sqrt{{(y_2-y_1)}^2+{(x_2-x_1)}^2}$

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.

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(x₁, y₂), midpoint M (x_m, y_m) and endpoint B (x₂, y₂), 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 x₁ or y₁ the subject of the formula. This means that the coordinate of the starting point in the x-axis, x₁ is:

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

and the coordinate of the starting point in the y-axis, y₁ is:

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

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

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

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