Division Of A Line Segment Into A Given Ratio

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.



Let use see how this method gives us the required division.

Since A₃C is parallel to A₅B, therefore,

( frac{A}}=frac{AC}{CB}text{ }left( text{By the Basic Proportionality Theorem} right) )

( frac{A}}=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.

[youtube https://www.youtube.com/watch?v=m4PSLIMfTiU?feature=oembed]

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.



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}{B}=frac{AC}{BC})

( frac{A}{B}=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.