Division Line Segments: Construction, Steps, Examples & Sample Questions

Division line segments are some of the most useful things to learn when it comes to segments and line segments in mathematics. Line segments can be divided into many parts. But these parts may or may not be equal. However, while performing the division of line segments, our focus is on dividing the line segments into equal parts. Let us learn more about division line segments with their construction steps and some solved examples.

Key takeaways: line segments, Division line segments, division of line segments, construction, dividing a line segment, dividing the line segment internally.

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What is a Line Segment?

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When some points or dots are connected along a straight path, it is known as a line. Moreover, it doesn't have endpoints and can be extended to infinity in both directions. So, a line is a 1-dimensional figure which has no starting and ending points. A line always has length but no width or height. Now, a line segment is a part of any line between both the endpoints of that line.

For example, if there is a straight line named CD and two points are marked in between that line as a part of the same then it is a line segment.

Therefore, division line segments are related to line segments. In other words, part of a line and not the entire line actually. A line segment can never be indefinite as being a part of a line, it has a starting and an ending point.
Read more: Line Segment and Ray

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What is Division of Line Segments?

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We can divide each and every line segment into 'n' number of equal parts. 'N' is a natural number there. So, as clear by the name itself, the division line segment is nothing other than dividing the line segment. For instance, one can divide a line segment of length 10 cm into two equal parts by using a ruler. The first step would be to mark a point 5 cm away from one end of the line segment. Therefore, 10 cm is divided into two 5 cm line segments. Likewise, if we have to divide a line segment of length 18 cm. It can be divided in the ratio of 2:1 as follow, CD is the line segment of length 18 cm. A is the point that divides the line segment in the ratio 2:1.

Let AC = x, then AD = 2x

(Given: AD+AC =18)

Putting the values in the formula,

AD + AC = 2x + x = 18, x = 6

AD = 12 cm and AC = 6 cm.
Mark point A, 12 cm away from C in the Line Segment.

So, that was about when we can measure the length of a line segment. But, what if one cannot measure the lengths precisely or at all? The point will not be marked correctly. No worries, there is a better way to mark a point while division line segments are done in a provided ratio. But, first, we will look at the internal and external division of the line segment.

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Internal and External Division Line Segments

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If a point performs the division of a line segment into two parts that may or may not be equal, one can use the section formula. This formula helps to find that point if the line segment's coordinates are known, and we can also use this formula to find the ratio in which the point divides a line segment only if the point's coordinates are provided. The use of this formula is determined by the position of the dividing point whether it is dividing the line segment internally or externally.

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Formula of Division of Line Segments

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Division of line segments can be done internally as well as externally. We have different formulas for both tasks.

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The formula of Internal Division of Line Segment

This formula is used for the internal division of line segments i.e. when the line segments are divided in the ratio of m:n internally. In this one, point C lies anywhere between points A and B. Therefore, the coordinates of point C will be,
(px₂+qx₁)/(p+q), (py₂+qy₁)/(p+q)

Here, X coordinates are,
(px₂ + qx₁)/ (p + q)

Y coordinates are,
(py₂ + qy₁) / (p + q)

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The Formula of External Division of Line Segment

This formula is used for external division of the line segments i.e. when we divide the line segments in the ratio of a: b externally. In this one, point P lies on the external parts of the line segment. Therefore, the coordinates of point P will be,

(ax₂+bx₁)/(a−b),(ay₂+by₁)/(a−b)

Here, X coordinates are,

(ax₂ - bx₁)/ (a - b)

and Y coordinates are,

(ay₂ - by₁) / (a - b)

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Construction Steps of Division of line Segments

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With the purpose of dividing a line segment internally in a given ratio m: n, where both m and n are positive integers,

We follow the following steps:

STEP OF CONSTRUCTION:

(i) Firstly, draw a line segment AB of a given length by using a line gauge or ruler.

(ii) Then, draw a ray AX that makes an acute angle with line segment AB.

(iii) Along AX mark off (m + n) points A1, A2,..., Am+n in a way so that AA1 = A1A2 = ....=Am+n+Am+n.

(iv) Join B to Am+n

(v) Now, draw a line parallel to Am+n B through the point Am by making an angle equal to the division of line segment Am.

Presume this line segment meets AB at a point P.

The point P therefore obtained is the required point that divides AB internally in the ratio m:n.

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Things to Remember

• A line segment can be divided into 'n' number of parts. 'N' will always be a natural number there.
• Division of line segments can be done internally as well as externally.
• The above-provided division of line segments in mathematics is especially helpful when the ratio is provided and we cannot measure the line segment precisely.

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