Before knowing about the division of a Line segment, let us know what a line and line segment means,

Line and line segment

A line is a collection of points along a straight path which extends to both the directions without endpoints.

AB is a line which doesn’t have any ending.

A line segment is a part of a line between two endpoints.

PQ is a line segment having P and Q as endpoints on the line AB.

Division of a line segment

A** line segment** can be divided into ‘n’ equal parts, where ‘n’ is any natural number.

For example; a line segment of length 10 cm is divided into two equal parts by using a ruler as,

- Mark a point 5 cm away from one end.
- 10 cm is divided into two 5 cm line segments.

Similarly, a line segment of length 15 cm can be divided in the ratio 2:1 as,

- AB is the line segment of length 15 cm and C divides the line in the ratio 2:1.
- Let CB = x, then AC = 2x
- AC + CB = 2x + x = 15, x = 5
- AC = 10 cm and CB = 5 cm.
- Mark point C, 10 cm away from A

Now, what if we cannot measure the lengths precisely? We will not be able to mark the point correctly.

There is a better way to mark point while dividing a line in a given ratio, which explained as follows.

Consider a line \( \overleftrightarrow{PQ}\) . We have to divide \( \overleftrightarrow{PQ}\) in a ratio m:n, where m and n are positive integers.

Let m = 3 and n = 1. So, we are dividing the line \( \overleftrightarrow{PQ}\) in the ratio 3 : 1.

Steps of construction

- Draw \(\overleftrightarrow{PQ}\) a ray PX which makes acute angle with \( \overleftrightarrow{PQ}\)
- Now, locate 4(m + n = 3 + 1) points A,B,C and D such that PA = AB = BC = CD using compass.
- Join Q and D using ruler.
- Draw a line through the point C (m = 3,C is the third point from P) which should be parallel to \( \overleftrightarrow{QD}\) by making an equal angle to ∠PDQ intersect the line PQ at R. (Refer figure.)
- Now, R is the point on \( \overleftrightarrow{PQ}\) ⃡which divides \( \overleftrightarrow{PQ}\) in the ratio 3:1.

Figure 1-Division of a line segment

Since CR is parallel to DQ;

By basic proportionality theorem,

\( \frac {PR}{RQ}\) = \( \frac {PC}{CD}\)

By construction, \( \frac {PC}{CD}\) = \( \frac {3}{1}\)

Therefore,

\( \frac {PR}{RQ}\) = \( \frac {3}{1}\)<

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