Ratios of Directed Line Segments Calculator

Created by Krishna Nelaturu
Reviewed by Steven Wooding
Last updated: Mar 21, 2022

The ratios of the directed line segments calculator will help you calculate the coordinates of the point that partition the line segment in a given proportion. This article will explore what a directed line segment is, how to partition a line segment with a given ratio with some examples, segment partition formula, and some frequently asked questions.

What is a directed line segment?

A line segment ABAB is a part of a line bound by two endpoints AA and BB where ABA \neq B. A directed line segment AB\overrightharpoon{AB} is a line segment with a definite direction – it is the line segment directed from AA to BB.

Difference between a line segment and a directed line segment.
A directed line segment has both length and direction. AB\overrightharpoon{AB} has the same length as the line segment ABAB, and is along the direction ABA\rightarrow B. Note that ABBA\overrightharpoon{AB} \neq \overrightharpoon{BA}.

While the line segment ABAB can also be written as BABA, the same is not true for the directed line segment AB\overrightharpoon{AB}. This is because AB\overrightharpoon{AB} is directed from AA to BB, whereas BA\overrightharpoon{BA} is directed from BB to AA.

🔎 Notice the similarities between a directed line segment and a vector? Not all directed line segments are vectors, but you can use a directed line segment to geometrically represent a vector with the same direction if the length of the line segment matches the magnitude of the vector.

In the next section, we shall answer that burning question in your mind: how do you divide a segment into given ratios?

Directed line segment partitioning and formula

A point PP lying on the directed line segment AB\overrightharpoon{AB} will divide into two line segments. There are two ways to divide a line segment:

  1. Internally, when the point PP lies somewhere within the segment AB\overrightharpoon{AB}; and
  2. Externally, when the point PP lies somewhere on the extended line segment AB\overrightharpoon{AB}.
Point P on AB divides AB in the ratio m:n.
Internal partition of AB\overrightharpoon{AB} into the ratio m:nm:n by a point P(px,py)P(p_x,p_y) lying on AB\overrightharpoon{AB}.

To partition the line segment AB\overrightharpoon{AB} internally into the ratio m:nm:n, the point P(px,py)P(p_x,p_y) must lie on AB\overrightharpoon{AB} such that it is mm+n\frac{m}{m+n} away from AA and nm+n\frac{n}{m+n} away from BB.

Point P on extended line segment AB divides it in the ratio m:n
External partition of AB\overrightharpoon{AB} into the ratio m:nm:n by a point P(px,py)P(p_x,p_y) lying on the extended line segmentAB\overrightharpoon{AB}.

On the other hand, to partition the line segment AB\overrightharpoon{AB} externally into the ratio m:nm:n, the point P(px,py)P(p_x,p_y) must lie on the extended line segment AB\overrightharpoon{AB} such that it is mmn\frac{m}{m-n} away from AA and nmn\frac{n}{m-n} away from BB.

Now that you know the concept of breaking a line segment into a ratio let's put together a formula for a directed line segment divided by any point PP.

For the internal partition of AB\overrightharpoon{AB}:

P(px,py)=(mx2+nx1m+n,my2+ny1m+n)\scriptsize P(p_x,p_y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)

And for the external partition of AB\overrightharpoon{AB}:

P(px,py)=(mx2nx1mn,my2ny1mn)\scriptsize P(p_x,p_y) = \left(\frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n}\right)

where:

  • PP – Any point that partitions the directed line segment AB\overrightharpoon{AB};
  • px,pyp_x,p_y – The x- and y-coordinates of the point PP;
  • m,nm,nRatio m:nm:n into which the point PP divides AB\overrightharpoon{AB};
  • x1,y1x_1,y_1 – The x- and y-coordinates of the endpoint AA of AB\overrightharpoon{AB}; and
  • x2,y2x_2,y_2 – The x- and y-coordinates of the endpoint BB of AB\overrightharpoon{AB}.

❗ Keep in mind that in case of external division of the line segment, the ratios mm and nn cannot be equal and must be distinct to avoid a division by zero in the formula.

Now that you know the ratio of line segments formula, let's discuss it further, along with some examples of segment partition calculation.

How do I partition a line segment with a given ratio?

To find the point P(pₓ, pᵧ) that internally divides the line segment AB into the ratio m:n, follow these steps:

  1. Calculate pₓ using pₓ = (mx₂ + nx₁)/(m + n), where x₁ and x₂ are the x-coordinates of A and B respectively.
  2. Determine pᵧ using pᵧ = (my₂ + ny₁)/(m + n), where y₁ and y₂ are the y-coordinates of A and B respectively.

To find the point P(pₓ, pᵧ) that externally divides the line segment AB into the ratio m:n, follow these steps:

  1. Compute pₓ using pₓ = (mx₂ - nx₁)/(m - n), where x₁ and x₂ are the x-coordinates of A and B respectively.
  2. Find pᵧ using pᵧ = (my₂ - ny₁)/(m - n), where y₁ and y₂ are the y-coordinates of A and B respectively.

For example, consider a line segment AB\overrightharpoon{AB} with the endpoints A(1,2)A(1,2) and B(4,6)B(4,6). The direction of this segment would be from AA to BB. To find a point that divides this segment internally in the ratio 2:32:3, we can use the internal partition formula as follows:

P(px,py)=(24+312+3,26+322+3)=(115,185)=(2.2,3.6)\scriptsize \begin{align*} P(p_x, p_y) &= \left(\frac{2 \cdot 4 + 3 \cdot 1}{2+3}, \frac{2 \cdot 6 + 3\cdot 2}{2+3}\right) \\\\ &= \left(\frac{11}{5}, \frac{18}{5}\right) \\\\ &= \left(2.2, 3.6\right) \end{align*}
Directed line segment AB divided into ratio 2:3 by a point P(2.2, 3.6).
Point P(2.2,3.6)P(2.2, 3.6) divides AB\overrightharpoon{AB} into the ratio 2:32:3.

Note that to say that the point P(2.2,3.6)P(2.2,3.6) divides AB\overrightharpoon{AB} in the ratio 2:32:3 is the same as saying that the point P(2.2,3.6)P(2.2,3.6) lies 25th{\frac{2}{5}}^{th} away from the endpoint A(1,2)A(1,2) and 35th{\frac{3}{5}}^{th} away from the endpoint B(4,6)B(4,6).

Now, if we want to divide the same line segment externally in the same ratio, then we shall employ the formula for external partition of the line segment:

P(px,py)=(243123,263223)=(831,1261)=(5,6)\scriptsize \begin{align*} P(p_x, p_y) &= \left(\frac{2 \cdot 4 - 3 \cdot 1}{2-3}, \frac{2 \cdot 6 - 3\cdot 2}{2-3}\right) \\\\ &= \left(\frac{8-3}{-1}, \frac{12-6}{-1}\right) \\\\ &= \left(-5,-6\right) \end{align*}
AB divided into ratio 2:3 by a point P(-5,-6) lying on extended AB.
Point P(5,6)P(-5,-6) divides AB\overrightharpoon{AB} into the ratio 2:32:3 externally.

See how easy it is to partition a line segment in a given ratio?😉 Go ahead and try some practice problems and master this method! You can always verify your results using this calculator to divide line segments.

How to use this ratios of directed line segments calculator

This ratios of directed line segments calculator is beneficial to find the point that divides a directed line segment in a given ratio or to find the ratio in which a given point splits the line segment.

  1. Choose the type of partition between internal and external in the The line is paritioned... field. By default, it is set to internal partition.

  2. Enter the coordinates of the endpoints of the segment. Ensure that you're getting the direction of the line segment correct. In this calculator, the direction is always from A(x1,y1)A(x_1,y_1) to B(x2,y2)B(x_2,y_2).

  3. This step depends on whether you want to find the point or the ratio:

    • If the ratio is the given value, enter the given ratio in the corresponding fields, and the coordinates of the point appear in their respective fields, along with a helpful graph. Otherwise, leave the ratio fields empty.
    • If the coordinates of the point are the given value, enter these coordinates in the corresponding fields, and the resulting ratio will appear at the bottom.

You now have a simple tool to help you calculate the partitioning of a line segment anytime you need! You can also use our other calculators to find out more interesting math:

FAQ

How do I find a point that divides a segment in half?

If you know the coordinates of the endpoints of the line segment, you can easily find its mid-point (xₘ, yₘ) using these steps:

  1. Calculate the average of the x-coordinates of the end-points to get the x-coordinate of the mid-point. xₘ = (x₁ + x₂)/2.
  2. Determine the average of the y-coordinates of the end-points to obtain the y-coordinate of the mid-point. yₘ = (y₁ + y₂)/2.
  3. Verify these results using our Midpoint Calculator or Ratios of Directed Line Segments Calculator.

How can I divide a line segment into three equal parts?

To divide a line segment AB into three equal parts, you need to find two points P(pₓ, pᵧ) and Q(Qₓ, Qᵧ) on AB, such that they each divide AB into the ratios 1:2 and 2:1:

  1. Calculate the x-coordinate pₓ of the point P using the formula pₓ = (2x₂ + x₁)/3, where x₁ and x₂ are the x-coordinates of A and B respectively.

  2. Compute the y-coordinate pᵧ of the point P using pᵧ = (2y₂ + y₁)/3, where y₁ and y₂ are the y-coordinates of A and B respectively.

  3. Determine the x-coordinate qₓ of the point Q using qₓ = (x₂ + 2x₁)/3.

  4. Find the y-coordinate qᵧ of the point Q using qᵧ = (y₂ + 2y₁)/3.

  5. Put these coordinates together to get the points P(pₓ, pᵧ) and Q(Qₓ, Qᵧ).

How to find a point lying one-third of the way from an endpoint?

A point P lying one-third of the way from the endpoint A on the line segment AB will divide it in the ratio 1:2. To find this point, follow these simple steps:

  1. Calculate the x-coordinate pₓ of this point using the formula pₓ = (2x₂ + x₁)/3, where x₁ and x₂ are the x-coordinates of A and B respectively.

  2. Compute the y-coordinate pᵧ of this point using pᵧ = (2y₂ + y₁)/3, where y₁ and y₂ are the y-coordinates of A and B respectively.

  3. Put them together to obtain the desired point P(pₓ, pᵧ).

Krishna Nelaturu
The line is partitioned...
internally.
Coordinates of line segment's end points
x₁
y₁
x₂
y₂
Ratio (m : n)
mᵢ
nᵢ
Coordinates of the point P(pₓ,pᵧ)
pₓᵢ
pᵧᵢ
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