**Proportionality Theorem**states that, A line parallel to one side of a triangle divides the other two sides into parts of equal proportion.

In triangle ABC, a line drawn parallel to BC cuts AB and AC at P and Q respectively.

Let the point P divide AB in the ratio of l: m where l and m are natural numbers. Divide AP into 'l' and PB into 'm' equal parts. Through each of these points on AB, draw lines parallel to BC to cut AC.

Basic Proportionality Theorem (B.P.T.) will be more useful in the topic 'SIMILARITY'.

Division of a line segment into equal parts.

Divide a line segment of length 8.4 cm into 5 equal parts.

AB = 8.4 cm

*SECOND METHOD*1. Draw AB = 8.4 cm and through A draw another line AX at an acute angle to AB.

2. With a suitable radius, cut off equal lengths AP, PQ, QR, RS and ST.

3. Join TB. Draw SF, RE, QD and PC parallel to TB to cut AB at F, E, D and C. The line segment AB is divided into five equal parts.

AC = CD = DE = EF = FB

Divide AB = 8.4 cm internally in the ratio of 3 : 2.

1. Draw AB = 8.4 cm and through A draw another line AX at an acute angle to AB.

2. Make ÐABY = ÐBAX so that BY is on the opposite side of AB to that of AX.

3. With suitable radius, cut off equal lengths AH, HJ, JK, KL and LM on AX.

Similarly, with the same radius cut off BP = PQ = QR = RS = ST on BY.

4. Join AT, HS, JR, KQ, LP and MB to cut AB at points C, D, E and F respectively.

AB is divided at E in the ratio of 3 : 2.

- Midpoint Theorem

- Converse of mid point theorem - The straight line drawn through the mid point of one side of a triangle, parallel to another, bisects the third side.
- The Intercepts Theorem

- A line parallel to one side of a triangle, divides the other two sides proportionally (Basic Proportionality Theorem).
- Using the above results, we can divide a line segment into a number of equal parts.

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