Division of Line Segments in Coordinate Geometry

The division of line segments is defined as a line that can be divided into n numbers 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 a line are:

• 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.

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Internal and External Division

If a point divides a line segment into two parts that may or may not be equal, we can use the section formula to find that point if the line segment's coordinates are known, and 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 obtain the coordinates of a point C that splits a line segment AB in the ratio m:n. There are two sorts of section formulas. These kinds are determined by the presence of point C, which can be found between the points or outside the line segment.

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 do 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.

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)

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.

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) intersecting 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 draw 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

Conclusion

The division of line segments is defined as a line that can be divided into n equal parts. There are two sorts of section formulas. We use the section formula to find the coordinates of a point C that splits a line segment AB in the ratio m:n.