Division of a Line Segment: Meaning, Different Methods, Problems

Division of a Line Segment: A point divides a line segment into two parts that may or may not be equal. We can use the section formula to find the coordinates of that point if the line segment’s coordinates are identified. We can also use the section formula to find the ratio in which the point divides the given line segment if the point’s coordinates are known.

We use the section formula to find the coordinates of a point \(C\) that divides a line segment \(AB\) in the ratio \(m:n\). A line segment can be divided in two ways: internally or externally. Let us learn the division of a line segment in detail.

Line and Line Segments

A line is a one-dimensional figure with no thickness. In geometry, a line can extend in both directions indefinitely. A line is a series of points connected in a single direction with no gaps between them.

A line segment is a section of a line bounded by two different ends and contains every point on the line in the shortest possible distance between them.

Division of Line Segment in a Given Ratio

Internal Division of a Line Segment: When a point divides a line segment in the ratio \(m:n\) internally at point \(M\), that point is in between the line segment’s coordinates.

Consider a line segment \(PQ\):

Suppose the given line is divided internally in the ratio \(3: 4\). This means that you have to find a point \(M\) on \(P Q\) such that \(P M: M Q=3: 4\), as shown below:

This is the internal division of a given line segment in a given ratio \(3: 4\)

External Division of a Line Segment: When the point which divides the line segment is divided externally in the ratio \(m: n\), the point which divides the line segment lies outside the line segment, i.e. when we extend the line, it coincides with the point.

Consider a line segment \(P Q\):

We want to find out a point lying on the extended line \(P Q\), outside of the segment \(P Q\), such that \(P M: Q M=3: 1\), as shown in the figure below:

We will say that \(M\) externally divides \(P Q\) in the ratio \(3: 1\)

Point Division of a Line Segment: The bisector is a line that divides the line or an angle into equal halves. The bisector of a segment always includes the midpoint of the segment. A line segment has two endpoints, i.e., it has a starting point and an ending point. A line segment bisector will be equidistant from these two endpoints.

Internal Division of a Line Segment

A formula is used to find the coordinates of the point of division when the segment is divided internally in the ratio \(m: n\). That is when point \(P\) is in the middle of the two points \(A\) and \(B\). This formula is used when the line segment is divided internally in the ratio \(m: n\). That is when point \(P\) is in the middle of two points \(A\) and \(B\).

For Internal division, the section formula is:

\(P(x, y)=\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}\right)\)

If we split, the \(x-\text {coordinate}\) and \(y-\text {coordinate}\) will get,

\(x-\text {coordinate} =\left(\frac{m x_{2}+n x_{1}}{m+n}\right), y- \text {coordinate}=\left(\frac{m y_{2}+n y_{1}}{m+n}\right)\)

External Division of a Line Segment

We use the section formula for the external division of the coordinates of the point \(C\) when it is on the external part of the line segment.

When you extend the segment outside its actual length, a point on the external part of the segment lies there, as shown in the figure above.

For external division, the section formula is:

\(C(x, y)=\left(\frac{m x_{2}-n x_{1}}{m-n}, \frac{m y_{2}-n y_{1}}{m-n}\right)\)

If we split, the \(x-\text {coordinate}\) and \(y-\text {coordinate}\) will be,

\(x- \text {coordinate} =\left(\frac{m x_{2}-n x_{1}}{m-n}\right), y- \text {coordinate} =\left(\frac{m y_{2}-n y_{1}}{m-n}\right)\)

Solved Examples – Division of a Line Segment

Q.1. Find the point coordinates that divide the line segment joining the \((4,6)\) and \((-5,-4)\) internally in the ratio \(2: 3\).
Ans: Let \(P(x, y)\) be the point that divides the line segment joining \(A(4,6)\) and \(B(-5,-4)\) internally in the ratio \(2: 3\).
Here,
\(\left(x_{1}, y_{1}\right)=(4,6)\)
\(\left(x_{2}, y_{2}\right)=(-5,-4)\)
\(m: n=2: 3\)
The section formula(internally) is given by,
\(P(x, y)=\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}\right)\)
\(P(x, y)=\left(\frac{2(-5)+3(4)}{2+3}, \frac{2(-4)+3(6)}{2+3}\right)\)
\(P(x, y)=\left(\frac{-10+12}{5}, \frac{-8+18}{5}\right)\)
\(P(x, y)=\left(\frac{2}{5}, \frac{10}{5}\right)\)
\(P(x, y)=\left(\frac{2}{5}, 2\right)\)
Therefore, \(x- \text {cordinate} =\frac{2}{5}, y- \text {cordinate} =2\)

Q.2. Find the point coordinates that divide the line segment joining the \((-1,7)\) and \((4,-3)\) internally in the ratio \(2: 3\).
Ans: Let \(P(x, y)\) be the point that divides the line segment joining \(A(-1,7)\) and \(B(4,-3)\) internally in the ratio \(2: 3\).
Here,
\(\left(x_{1}, y_{1}\right)=(-1,7)\)
\(\left(x_{2}, y_{2}\right)=(4,-3)\)
\(m: n=2: 3\)
The section formula(internally) is given by,
\(P(x, y)=\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}\right)\)
\(P(x, y)=\left(\frac{2(4)+3(-1)}{2+3}, \frac{2(-3)+3(7)}{2+3}\right)\)
\(P(x, y)=\left(\frac{8-3}{5}, \frac{-6+21}{5}\right)\)
\(P(x, y)=\left(\frac{5}{5}, \frac{15}{5}\right)\)
\(P(x, y)=(1,3)\)
Therefore, \(x- \text {cordinate} =\frac{2}{5}, y- \text {cordinate} =3\)

Q.3. In what ratio does the point \((-4,6)\) divides the line segment joining the points \(A(-6,10)\) and \(B(3,-8)\) ?
Ans: Let \((-4,6)\) divide \(A B\) internally in the ratio \(m: n\).
Use section formula,
\(P(x, y)=\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}\right)\)
\((-4,6)=\left(\frac{3 m-6 n}{m+n}, \frac{8 m+10 n}{m+n}\right)\)
\(\Rightarrow\left(\frac{3 m-6 n}{m+n}\right)=-4\)
\(\Rightarrow-4 m-4 n=3 m-6 n\)
\(\Rightarrow-7 m=-2 n\)
\(\Rightarrow 7 m=2 n\)
\(\Rightarrow m: n=2: 7\)
Therefore, the point \((-4,6)\) divides the line segment joining the points \(A(-6,10)\) and \(B(3,-8)\) in the ratio \(2: 7\).

Q.4. Find the point coordinates that divide the line segment joining the \((-1,2)\) and \((4,-5)\) externally in the ratio \(3: 2\).
Ans: Let \(P(x, y)\) be the point that divides the line segment joining \(A(-1,2)\) and \(B(4,-5)\) internally in the ratio \(3: 2\).
Here,
\(\left(x_{1}, y_{1}\right)=(-1,2)\)
\(\left(x_{2}, y_{2}\right)=(4,-5)\)
\(m: n=3: 2\)
The section formula(externally) is given by,
\(P(x, y)=\left(\frac{m x_{2}-n x_{1}}{m-n}, \frac{m y_{2}-n y_{1}}{m-n}\right)\)
\(P(x, y)=\left(\frac{3(4)-2(-1)}{3-2}, \frac{3(-5)-2(2)}{3-2}\right)\)
\(P(x, y)=\left(\frac{12+2}{1}, \frac{-15-4}{1}\right)\)
\(P(x, y)=(14,-19)\)
Therefore, \(x-\text {cordinate} =14, y- \text {cordinate} =-19\)

Q.5. Find the point coordinates that divide the line segment joining the \((3,2)\) and \((6,5)\) externally in the ratio \(2: 1\).
Ans: Let \(P(x, y)\) be the point that divides the line segment joining \(A(3,2)\) and \(B(6,5)\) externally in the ratio \(2: 1\).
Here,
\(\left(x_{1}, y_{1}\right)=(3,2)\)
\(\left(x_{2}, y_{2}\right)=(6,5)\)
\(m: n=2: 1\)
The section formula(externally) is given by,
\(P(x, y)=\left(\frac{m x_{2}-n x_{1}}{m-n}, \frac{m y_{2}-n y_{1}}{m-n}\right)\)
\(P(x, y)=\left(\frac{2(6)-1(3)}{2-1}, \frac{2(5)-1(2)}{2-1}\right)\)
\(P(x, y)=\left(\frac{12-3}{1}, \frac{10-2}{1}\right)\)
\(P(x, y)=(9,8)\)
Therefore, \(x- \text {cordinate} =9, y- \text {cordinate} =8\)

Q.6. In what ratio does the point \((-1,6)\) divides the line segment joining the points \(A(-3,10)\) and \(B(6,-8)\) ?
Ans: Let \((-1,6)\) divide \(A B\) internally in the ratio \(m: n\).
Use section formula,
\(P(x, y)=\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}\right)\)
\((-1,6)=\left(\frac{6 m-3 n}{m+n}, \frac{-8 m+6 n}{m+n}\right)\)
\(\Rightarrow\left(\frac{6 m-3 n}{m+n}\right)=-1\)
\(\Rightarrow-m-n=6 m-3 n\)
\(\Rightarrow-7 m=-2 n\)
\(\Rightarrow 7 m=2 n\)
\(\Rightarrow m: n=2: 7\)
Therefore, the point \((-1,6)\) divides the line segment joining the points \(A(-3,10)\) and \(B(6,-8)\) in the ratio \(2: 7\).

Q.7. Find the point coordinates that divide the line segment joining the \((4,6)\) and \((-5,-4)\) internally in the ratio \(3: 4\)
Ans: Let \(P(x, y)\) be the point that divides the line segment joining \(A(4,6)\) and \(B(-5,-4)\) internally in the ratio \(3: 4\).
Here,
\(\left(x_{1}, y_{1}\right)=(4,6)\)
\(\left(x_{2}, y_{2}\right)=(-5,-4)\)
\(m: n=3: 4\)
The section formula(internally) is given by,
\(P(x, y)=\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}\right)\)
\(P(x, y)=\left(\frac{3(-5)+4(4)}{3+4}, \frac{3(-4)+4(6)}{3+4}\right)\)
\(P(x, y)=\left(\frac{-15+16}{7}, \frac{-12+24}{7}\right)\)
\(P(x, y)=\left(\frac{1}{7}, \frac{12}{7}\right)\)
Therefore, \(x- \text {cordinate} =\frac{1}{7}, y- \text {cordinate} =\frac{12}{7}\)

Summary

A line segment is a section bounded by two different ends and contains every point in the shortest possible distance between them. When a point divides a line segment in the ratio \(m:n\) internally at a point, that point is in between the line segment’s coordinates. When the point which divides the line segment is divided externally in the ratio \(m:n\), the point which divides the line segment lies outside the line segment, i.e. when we extend the line, it coincides with the point. This article includes the internal and external division of a line segment, section formulas, and problems.

This article, “Division of a Line Segment,” help in understanding these in detail, and it helps solve the problems based on these very easily.

Frequently Asked Questions (FAQs)

Q.1. Define line segment.
Ans: A line segment is a section bounded by two different ends and contains every point in the shortest possible distance between them.

Q.2. What is the point of division of a line segment?
Ans: A line segment has two endpoints: one at the starting and one at the ending. A point of division of a line segment is a point that divides the line segment into parts that may be equal or unequal.

Q.3. How do you divide a segment?
Ans: We can divide a segment into two equal parts if we can find its midpoint. We may find the points needed to divide the entire section into four equal parts by finding the midpoints of these two equal parts. If the midpoints of each of these segments are found, then the segment can be divided into eight equal parts, and so on.

Q.4. What is a line division?
Ans: A line is made up of infinite points extending in either direction. A line segment is a part of a line with two fixed endpoints. A line division means dividing the line into two or more parts. Any natural number \(n\) could divide a line segment into ‘\(n\)’ equal parts. 
For example, \(8 \,\text {cm}\) long line segment could be divided into two equal parts by drawing a point \(4 \,\text {cm}\) away from one end with such a ruler.

Q.5. What is it called when points lie on the same line?
Ans: Points that lie on the same line are called collinear points.

Q.6. What is meant by division line segment point?
Ans: A division line segment point is a point that makes it possible to divide a line segment in a ratio.

Q.7. What is the internal division of the line segment?
Ans: When a point divides a line segment in the ratio m:n internally at point \(C\), that point is in between the line segment’s coordinates.

Q.8. What is the external division of the line segment?
Ans: When the point which divides the line segment is divided externally in the ratio \(m:n\), the point which divides the line segment lies outside the line segment, i.e. when we extend the line, it coincides with the point.

We hope this detailed article on the division of a line segment helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!