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| Basic Proportionality Theorem |
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| A line parallel to one side of a triangle divides the other two sides into parts of equal proportion. |
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| In triangle ABC, a line drawn parallel to BC cuts AB and AC at P and Q respectively. |
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| 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. |
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| Basic Proportionality Theorem (B.P.T.) will be more useful in the topic 'SIMILARITY'. |
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| Division of a line segment into equal parts. |
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| Divide a line segment of length 8.4 cm into 5 equal parts. |
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| AB = 8.4 cm |
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| 1. Draw AB = 8.4 cm and through A draw another line AX at an acute angle to AB. |
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| 2. With a suitable radius, cut off equal lengths AP, PQ, QR, RS and ST. |
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| 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. |
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| AC = CD = DE = EF = FB |
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| Second Method: |
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| Divide AB = 8.4 cm internally in the ratio of 3 : 2. |
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| 1. Draw AB = 8.4 cm and through A draw another line AX at an acute angle to AB. |
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| 2. Make ÐABY =
ÐBAX so that BY is on the opposite side of AB to that of AX. |
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| 3. With suitable radius, cut off equal lengths AH, HJ, JK, KL and LM on AX. |
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| Similarly, with the same radius cut off BP = PQ = QR = RS = ST on BY. |
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| 4. Join AT, HS, JR, KQ, LP and MB to cut AB at points C, D, E and F respectively. |
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| AB is divided at E in the ratio of 3 : 2. |
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