prove that If a line parallel to a side of a traingle intersect the remaining sides in two distinct point then the line divides the sides in the same proportion.
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Heya ,
Given That :
In ∆ABC the line l || side BC line I intersecet
Side AB & side AC in P & Q respectively ( refer the diagram )
To Prove :
AP / PB = AQ / QC
Construction :
Draw a segment PC & segment QB .
Proof :
A ( ∆APQ ) / A ( ∆PBQ ) = AP / PB ┈┈ (Ⅰ)
(Areas that are proportional to the bases)
A (∆PAQ) / A(∆PQB) = AQ / QC ┈┈ (Ⅱ)
(Areas in porportion to the bases)
∆PBQ & ∆ PQC having the same bases PQ & PQ II BC , their height same .
A(∆ PQB ) = A ( ∆ PQC ) ┈┈ (Ⅲ)
∴A(∆APQ) / A(∆PQB) = A(∆APQ) / A(∆PQC) ┈┈ From (Ⅰ), (Ⅱ) & (Ⅲ)
∴ AP / PB = AQ / QC ┈┈ From (I) , (II)
Hence , Proved
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