How to divide a line segment in a given ratio?

In this blog, we will introduce how to divide a line segment. Suppose a line segment is given and you have to divide it in a given ratio, say 3:2. You may do it by measuring the length and then marking a point on it that divides it in the given ratio. But suppose you do not have any way of measuring it precisely, how would you find the point. Let us see how to divide a line segment in a given ratio.

Divide A Line Segment In a Given Ratio

Construction: To divide a line segment in 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 make it clear, 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 that is  ( m + n) points A₁, A₂, A₃, A₄and 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 point C. Then, AC: CB = 3:2.

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

Since A3C is parallel to A5B, therefore, AA₃/A₃A₅ = AC/CB (By the Basic Proportionality Theorem) By construction AA₃/A₃A₅ = 3 by2 this implies AC/CB = 3 by 2. This shows that C divides AB in the ratio 3: 2.

Now let us construct a triangle similar to a given triangle whose sides are in a given ratio to the corresponding sides of the given triangle.

Construction: To construct a triangle similar to a given triangle as per the given scale factor. This construction involves two different situations. In one, the triangle to be constructed is smaller and in the other, it is larger than the given triangle.

Here, the scale factor means the ratio of the sides of the triangle to be constructed with the corresponding sides of the given triangle. Let us take these examples for understanding the constructions involved.

Example: Construct a triangle similar to a given triangle ABC with its sides equal to 3/4 of the corresponding sides of the triangle ABC (i.e., of scale factor 3/4).

Solution: Given a triangle ABC, we are required to construct another triangle whose sides are of the corresponding sides of the triangle ABC.

Steps of Construction :

1. Draw any ray BX making an acute angle with BC on the side opposite to the vertex A.

2. Locate 4 (the greater of 3 and 4 in 3/4) points B1, B₂, B3 and B4 on BX so that BB1 = B1B_? = B₂B3 = B3B4.

3. Join B₄C and draw a line through B₃ parallel to B4C to intersect BC at C dash.

4. Draw a line through C dash parallel to the line CA to intersect BA at A dash. Then, ?ABC is the required triangle.

Example: Construct a triangle similar to a given triangle ABC with its sides equal to 5/3 of the corresponding sides of the triangle ABC (i.e., of scale factor 5/3).

Solution: Given a triangle ABC, we are required to construct a triangle whose sides are5/3 of the corresponding sides of ?ABC.

Steps of Construction :
1. Draw any ray BX making an acute angle with BC on the side opposite to the vertex A.
2. Locate 5 points (the greater of 5 and 3 in 5/3) B1, B2, B3, B4 and B5 on BX so that BB1 = B1B2 = B2B3 = B3B4 = B4B5.
3. Join B3 (the 3rd point, 3 being smaller of 3 and 5 in 5/3 ) to C and draw a line through B5 parallel to B3C, intersecting the extended line segment BC at C dash.
4. Draw a line through C dash parallel to CA intersecting the extended line segment BA at A dash.
Then ABC is the required triangle.

Read More: Construct a Tangent to a Circle: Step Of Construction - Studynlearn