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Answer» If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.Basic Proportionality Theorem ProofLet us now try to prove the basic proportionality theorem statementConsider a triangle ΔABC, as shown in the given figure. In this triangle, we draw a line PQ parallel to the side BC of ΔABC and intersecting the sides AB and AC in P and Q, respectively.According to the basic proportionality theorem as stated above, we need to prove:AP/PB\xa0=\xa0AQ/QCConstructionJoin the vertex B of ΔABC to Q and the vertex C to P to form the lines BQ and CP and then drop a perpendicular QN to the side AB and also draw PM⊥AC as shown in the given figure.ProofNow the area of ∆APQ = 1/2 × AP × QN (Since, area of a triangle= 1/2× Base × Height)Similarly, area of ∆PBQ=\xa01/2\xa0× PB × QNarea of ∆APQ =\xa01/2\xa0× AQ × PMAlso,area of ∆QCP =\xa01/2\xa0× QC × PM ………… (1)Now, if we find the ratio of the area of triangles ∆APQand ∆PBQ, we havearea\xa0of\xa0ΔAPQarea\xa0of\xa0ΔPBQ\xa0=\xa012\xa0×\xa0AP\xa0×\xa0QN12\xa0×\xa0PB\xa0×\xa0QN\xa0=\xa0APPBSimilarly,\xa0area\xa0of\xa0ΔAPQarea\xa0of\xa0ΔQCP\xa0=\xa012\xa0×\xa0AQ\xa0×\xa0PM12\xa0×\xa0QC\xa0×\xa0PM\xa0=\xa0AQQC\xa0………..(2)According to the property of triangles, the triangles drawn between the same parallel lines and on the same base have equal areas.Therefore, we can say that ∆PBQ and QCP have the same area.area of ∆PBQ = area of ∆QCP …………..(3)Therefore, from the equations (1), (2) and (3) we can say that,AP/PB = AQ/QCAlso, ∆ABC and ∆APQ fulfil the conditions for similar triangles, as stated above. Thus, we can say that ∆ABC ~∆APQ.The MidPoint theorem is a special case of the basic proportionality theorem.According to mid-point theorem, a line drawn joining the midpoints of the two sides of a triangle is parallel to the third side.Consider an ∆ABC.ConclusionWe arrive at the following conclusions from the above theorem:If P and Q are the mid-points of AB and AC, then PQ || BC. We can state this mathematically as follows:If P and Q are points on AB and AC such that AP = PB = 1/2 (AB) and AQ = QC = 1/2 (AC), then PQ || BC.Also, the converse of mid-point theorem is also true which states that the line drawn through the mid-point of a side of a triangle which is parallel to another side, bisects the third side of the triangle.Hence, the basic proportionality theorem is proved.
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