# Divide a line seg

We need to divide this line segment AB of length 9 cm internally in the ratio 4 : 3.

Step 1: Draw a line segment AC of arbitrary length and at an any angle to AB such that CAB is acute.

Step 2: We plot (4 + 3 =) 7 points A1, A2, A3, A4, A5, A6, and A7 such that AA1 = A1A2 = A2A3 = A3A4 = A4A5 = A5A6 = A6A7

Step 3: We join points A7 and B.

Step 4: We draw line segment A4P such that A4P || A7B and P is the point of intersection of this line segment with AB.

Point P divides AB in the ratio 4 : 3.

Justification

In ΔAA4P and ΔAA7B,

iii. A is common.

iv. AA4P = AA7B (corresponding angles A4P || A7B)

Hence, ΔAA4P ~ ΔAA7B

So, ratio of lengths of corresponding sides must be equal.

Let AA1 = A1A2 = A2A3 = A3A4 = A4A5 = A5A6 = A6A7 = x

So, the previous relation can be re – written as –

4(AP +PB) = 7AP

4PB = 3AP

AP/PB = 4/3, or, AP : PB = 4 : 3

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