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Division Of A Line Segment

Constructions of Class 10

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:

STEP OF CONSTRUCTION:

(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 A1, A2,..., Am+n such that AA1 = A1A2 = ....= Am+n+Am+n.

(iv) Join B Am+n

(v) Through the point Am draw a line parallel to Am+n B by making an angle equal to ∠AAm + 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.

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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  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) intersecting AB at the point C

Division Of A Line Segment

Then, AC : CB = 3 : 2

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

Since A3C is parallel to A5B therefore,

Division Of A Line Segment (By the basic proportionality theorem)

By construction, Division Of A Line Segment Therefore, Division Of A Line Segment.

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 B1,B2,B3 (number of parts should be larger of the 2 and 3 in 2/3) such that BB1 = B1B2 = B2B3.

3. Join B3C and draw a line through B2 (the second point) parallel to B3C which meets BC at C′.

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

Division Of A Line Segment

Do solve NCERT text book for class 10 maths with the help of Physics Wallah NCERT solutions for class 10 maths. 

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