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Question
In a triangle ABC, line l || Side BC and line l intersects side AB and AC in points P and Q, respectively. Prove that: `"AP"/"BP"="AQ"/"QC"`
Solution
Construction: Join seg PC and seg BQ.
Given: In ΔABC , line l || BC
Line l intersects side AB and side AC in points P and Q, respectively.
A-P-B and A-Q-C
`"To Prove: ""AP"/"BP"="AQ"/"QC"`
Construction: Join seg BQ and seg CP
In ΔAPQ and ΔBPQ ,
`(A(triangleAPQ))/(A(triangleBPQ))=(AP)/(BP) " (i) ...........""(Triangles having equal height)"`
In ΔAPQ and ΔCPQ
`(A(triangleAPQ))/(A(triangleCPQ))=(AQ)/(CQ)" (ii) ......(Triangles having equal height)"`
`A(triangleBPQ)=A(triangleCPQ)" (iii) .....(Triangles with common base PQ and same height)"`
`(A(triangleAPQ))/(A(triangleBPQ))=(A(triangleAPQ))/(A(triangleCPQ))" ..............From (i), (ii) and (iii)"`
`"AP"/"BP"="AQ"/"QC"`
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