Division Of Line Segment

Division of Line Segment Definition

The division of line segments is defined as a line that can be divided into n number of equal parts where “n’ is determined as any natural number. Before discussing briefly the division of line segment, division of line segment formula, division of line segment example, we will first learn what is line and line segment.


Line

A line is defined as the straight- set points that can be extended indefinitely in both directions.

Some characteristics of the  line

  • It is a one-dimensional figure

  • It has no starting and ending points 

  • It has length, but no width or height 

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Line Segment

The line segment is a part of a line that cannot be extended indefinitely in both directions as it has both a starting point and an ending point. Some geometrical figures such  as triangle, polygon, hexagon are made from the line segment.(image will be uploaded soon)


Division of the Line Segment in a Given Ratio

As we know, the line segment can be divided into n equal parts where “n” is considered as any natural number.


For example- A line segment of 10 cm is divided into two equal halves i.e. through a ruler as,

  • Specify a point 5cm away from one end

  • 10 cm will be divided into two 5cm line segments


Here, we will  divide a line segment of 15 cm in the ratio of 2:1 as,

  • Let us take CB = x, then AC will be 2x

  • AC + CB = 2x + x = 15, x = 5

  • AC = 10 cm

  • CB = 5 cm

Mark a point C, 10 cm away from the point A.

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Now, we will not be able to mark the points accurately if we will not measure the length precisely.


The more accurate way to mark points while dividing a line in a given ratio is explained below.


Take a line segment PQ. We have to divide \[\overline{PQ}\] in the ratio of x:y , where x and y are considered as positive integers.


Let x = 3 and y =1. So, we are dividing the \[\overline{PQ}\] in the ratio of 3:1.


Division of line construction steps

  1. Draw a line segment PQ and a ray PX making an acute angle with the previously drawn line segment PQ

  2. As the given ratio is 3:1, the number of points to be specified on the ray PX should be 4 because (x + y= 3+1).

  3. With the help of the compass, mark the points A, B, C and D such that PA=AB=AC=AD

  4. Now, draw an arc from the point P with a measure of n/4 and mark the point as A. Draw another arc by taking the center A (first marked arc) with the same measure and name it as B. Similarly, draw the other two arcs also. For example, if the measure of the line segment PQ = 20cm, then divide 20 by 4, the result will be 5). Now measure 5 cm through the compass. Taking P as the center, draw an arc on the ray PX and mark it as a point A. Now draw another arc on the ray PX, taking A as a center and mark the point as B. Similarly, repeat the steps and draw 2 more arcs and mark them as C and D respectively.

  5. Join Q and D with the help of a ruler.

  6. Draw a line from point C (x =3) and it should be parallel to line segment QD by making an equal angle to ∠PDQ intersecting \[\overline{PQ}\] at R. Now, R is the point on line segment PQ which is dividing the \[\overline{PQ}\] in the ratio 0f 3:1.

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Now, let us see the way this method provides the required division.

As CR is parallel to DQ

Through the basic proportionality theorem, we get

PR/RQ = PC/CD

By construction, we get

PC/CD = 3/1

Hence,

PR/RQ = 3/1


Division of line segments formulas

The internal division of the line segment formula

The following formula is used when the line segments are divided in the ratio of p: q internally.

Here, the point C lies anywhere between the points A and B.

The coordinates of point C will be,

\[\frac{(px_{2} + qx_{1})}{(p + q)}\] , \[\frac{(py_{2} + qy_{1})}{(p + q)}\] 

X coordinates are

(px2 + qx1)/ (p + q)

Y coordinates are

 (py2 + qy1) / (p + q)

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The external division of the line segment formula

The following formula is used when the line segments are divided in the ratio of a: b externally

Here, the point P lies on the external parts of line segment

The coordinates of point P will be,

\[\frac{(ax_{2} + bx_{1})}{(a - b)}\] , \[\frac{(ay_{2} + by_{1})}{(a - b)}\] 

X coordinates are

(ax2 - bx1)/ (a - b)

Y coordinates are

(ay2 - by1) / (a - b)

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Solved Examples

 Here are some of the division of line segment examples which will clear your concepts

1. Construct a line segment in the ratio of 3:2.

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Solution: Given the line segment PQ, we want to divide \[\overline{PQ}\] in the ratio of 3:2

Here are the steps to draw a line segment PQ in the ratio of 3:2

1. Draw a ray PX which makes an acute angle with PQ.

2.  Locate 5 = ( m + n) points p1 ,p2, p3, p4, and p5 on PX such that

PP₁ = P1P2 =P2P3= P3P4 =P4P5

3. Join QP5

4. With the point, P3(m=3) draws a parallel line to P5Q (by making an acute angle to PP5 B) intersect PQ at the point R.

Then, PR: RQ = 3:2

Now, let us see the way this method provides the required division

As, P3Q is parallel to P5Q, therefore

By the basic proportionality theorem we get,

P P3/ P3P5 = PR/RQ

By construction, P P3/ P3P5 = 3/2

Hence, PR/RQ = 3/2


2. Construct a line segment of 7.6 cm and divide it in the ratio of 5:8. Measure it in two parts and write the steps of construction.

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Solution: Steps of construction of line segment 7.6 cm which is divided in the ratio of 5:8 are as follows:

1. Draw a line segment of length 7.6 cm and also drew a ray AX making an acute angle with line segment AB drawn previously.

2. Mention 13 points ( 5 + 8)  on ray AX as A1, A2, A3, A4,A5,A6…………A13 such that AA₁= A₁A₂= A₂A₃ and so on.

3. Draw a parallel line to BA13 through point A5 (by making an angle equal to AA13B) at the A5 intersecting line segment AB at point C.

4. Measure the length of the AC and AB

5. It will come out AB = 2.9 cm and CB= 4.7 cm


Quiz Time

1. When a line segment is divided in the ratio of 2:3, how many parts is it divided into?

a. 2/3

b. 2

c. 3

d. 5


2. What will be the length of the line segment if the coordinates A and B are (2,2) and (9,11) respectively?

a. 11.4

b. 13.4

c. 15.4

d. 17.4


3. A line segment can be drawn by joining

a. Two points

b. Three points

c. Four points

d. More than 3 points

FAQ (Frequently Asked Questions)

1. Explain Line Segment Bisector.

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 divides the line segment into two equal parts. A line segment bisector passes through the midpoint of the line segment. In the figure given below, PQ is the bisector of the line segment AB.


Example- Let us take a line segment AB = 4cm. A line segment bisector will divide the line segment into two equal halves of 2 cm each. If the bisector divides the line into two equal halves at 90°, then the bisector will be considered as a perpendicular bisector.

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2. How to Construct a Line Segment Using a Ruler and Compass?

Here we will learn the steps to draw a line segment through the compass and measuring ruler or scale.


Suppose we need to draw a line segment of length 7cm. Here are the steps, which we have to follow.

  1. The first step is to draw a line of any length. Mark a point X on the line, which will be considered as the starting point of the line segment.

  2. Using a ruler, locate the pointer of the compass 7cm away from the pencil lead,

  3.  Now, place a pointer of the compass at X and draw an arc on the line with the pencil point.

  4. Mark a point where both the line and arc intersect as Y.

  5. XY will be the required line segment of length 7 cm.