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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 - Geometry

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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"`

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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"`

 

Concept: Similarity of Triangles
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