# Determine a point We need to divide this line segment AB of length 12 cm internally in the ratio 2 : 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 (2 + 3 =) 5 points A1, A2, A3, A4, and A5 such that AA1 = A1A2 = A2A3 = A3A4 = A4A5. Step 3: We join points A5 and B. Step 4: We draw line segment A2P such that A2P || A5B and P is the point of intersection of this line segment with AB.

Point P divides AB in the ratio 2 : 3.

Justification

In ΔAA2P and ΔAA5B,

i. A is common.

ii. AA2P = AA5B (corresponding angles A2P || A5B)

Hence, ΔAA2P ~ ΔAA5B

So, ratio of lengths of corresponding sides must be equal. Let AA1 = A1A2 = A2A3 = A3A4 = A4A5 = x

So, the previous relation can be re – written as – 2(AP +PB) = 5AP

2PB = 3AP

AP/PB = 2/3, or, AP : PB = 2 : 3

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