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Division Of A Line Segment Into A Given Ratio

December 21, 2020 by Veerendra

Division Of A Line Segment Into A Given Ratio

Given a line segment AB, we want to divide it in the ratio m : n, where both m and n are positive integers. To help you to understand it, we shall take m = 3 and n = 2.
Steps of Construction:
1.  Draw any ray AX, making an acute angle with AB.
2.  Locate 5(= m + n) points A1, A2,  A3, A4 and A5 on AX so that AA1 = A1A2 = A2A3 = A3A4 = A4A5.
3.  Join BA5.
4.  Through the point A3 (m = 3), draw a line parallel to A5B (by making an angle equal to ∠AA5B) at A3 intersecting AB at the point C (see figure). Then, AC : CB = 3 : 2.
Division Of A Line Segment Into A Given Ratio 1
Let use see how this method gives us the required division.
Since A3C is parallel to A5B, therefore,
\( \frac{A{{A}_{3}}}{{{A}_{3}}{{A}_{5}}}=\frac{AC}{CB}\text{ }\left( \text{By the Basic Proportionality Theorem} \right) \)
\( \frac{A{{A}_{3}}}{{{A}_{3}}{{A}_{5}}}=\frac{3}{2}\text{ (By construction) } \)
\( \text{ }\frac{AC}{CB}=\frac{3}{2}\text{ } \)
This shows that C divides AB in the ratio 3 : 2.


Alternative Method
Steps of Construction :
1.  Draw any ray AX making an acute angle with AB.
2.  Draw a ray BY parallel to AX by making ∠ABY equal to ∠BAX.
3.  Locate the points A1, A2, A3 (m = 3) on AX and B1, B2 (n = 2) on BY such that AA1 = A1A2 = A2A3 = BB1 = B1B2.
4.  Join A3B2.
Division Of A Line Segment Into A Given Ratio 2
Let it in intersect AB at a point C (see figure)
Then AC : CB = 3 : 2
Whey does this method work ? Let us see.
Here DAA3C is similar to DAB2C. (Why ?)
\( \text{Then }\frac{A{{A}_{3}}}{B{{B}_{2}}}=\frac{AC}{BC}\)
\( \frac{A{{A}_{3}}}{B{{B}_{2}}}=\frac{3}{2}\text{ (By construction) } \)
\( \text{ }\frac{AC}{BC}=\frac{3}{2} \)
In fact, the methods given above work for dividing the line segment in any ratio.
We now use the idea of the construction above for constructing a triangle similar to a given triangle whose sides are in a given ratio with the corresponding sides of the given triangle.

 

Filed Under: Mathematics Tagged With: Constructions, Division Of A Line Segment, How To Divide A Line Segment

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