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Basic Proportionality Theorem

Statement: 

If a line is drawn parallel to the one side of the triangle to intersect other two sides of the triangle in distinct points, then other two sides are divided in the same ratio.

Given: 

ΔABC,PQ||BC

To Prove: 

APBP=AQCQ

Construction: 

QMAB,PNAC, Join P & C and B & Q.

Proof: 

Two triangles on the same base and between the same parallels are equal in area.

So, ar(ΔBPQ)=ar(ΔCPQ)          (1)

Area of triangle =12× Base × Height

ar(ΔAPQ)=12×AP×MQ        (2)

ar(ΔBPQ)=12×BP×MQ        (3)

ar(ΔAPQ)=12×AQ×PN        (4)

ar(ΔCPQ)=12×CQ×PN         (5)

From (1)(2), (3)(4) and (5), we get

ar(ΔAPQ)ar(ΔBPQ)=ar(ΔAPQ)ar(ΔCPQ)

12×AP×MQ12×BP×MQ=12×AQ×PN12×CQ×PN

APBP=AQCQ 

Hence Proved.

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