In order to divide a line segment internally is a given ratio m: n, where both m and n are positive integers,

We follow the following steps:

(i) Draw a line segment AB of given length by using a ruler.

(ii) Draw and ray AX making an acute angle with AB.

(iii) Along AX mark off (m + n) points A_{1}, A_{2},..., Am+n such that AA_{1} = A_{1}A_{2} = ....= A_{m+n}+A_{m+n}.

(iv) Join B A_{m+n}

(v) Through the point Am draw a line parallel to Am+n B by making an angle equal to ∠AA_{m + n} Bat Am.

Suppose this line meets AB at a point P.

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

**question **1. To divide a line segment in a given ratio 3 : 2.

**Solution: ** Given a line segment AB, we want to divide it in the ratio 3 : 2.

Steps of construction:

1. Draw any ray AX, making an acute angle with AB.

2. Locate 5(= m + n) points A_{1},A_{2},A_{3},A_{4,} and A_{5} on AX so that

AA1 = A_{1}A_{2} = A_{2}A_{3} = A_{3}A_{4} = A_{4}A_{5}.

3. Join BA_{5}.

4. Through the point A_{3} (m = 3), draw a line parallel to A_{5}B (by making an angle equal to ∠AA_{5}B) intersecting AB at the point C

Then, AC : CB = 3 : 2

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

Since A_{3}C is parallel to A_{5}B therefore,

(By the basic proportionality theorem)

By construction, Therefore, .

This shows that C divides AB in the ratio 3 : 2

We now use the above ideas of construction for constructing a triangle similar to a given triangle above whose sides are in a given ratio with the corresponding sides of the given triangle.

**Question **2. Construct a triangle of sides 4 cm, 5 cm and 6 cm then construct a triangle similar to it whose sides are 2/3 of the corresponding sides of it.

**Solution: ** Steps of construction:

1. Take BC = 6 cm. Let ∠CBX be any acute angle (< 90°).

2. Mark the three points B_{1},B_{2},B_{3} (number of parts should be larger of the 2 and 3 in 2/3) such that BB_{1} = B_{1}B_{2} = B_{2}B_{3.}

3. Join B_{3}C and draw a line through B_{2} (the second point) parallel to B_{3}C which meets BC at C′.

4. Draw C'A' parallel to CA which intersects AB at A′.

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