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In a ΔABC, BM and CN are perpendiculars from B and C respectively on any line passing through A. If L is the mid-point of BC, prove that ML = NL.



Given: l is a straight line passing through the vertex A of ΔABC. BM ⊥ l and CN ⊥ l. L is the mid point of BC.To prove: LM = LNConstruction: Draw OL ⊥ lProof:If a transversal make equal intercepts on three or more parallel line, then any other transversal intersecting then will also make equal intercepts.BM ⊥ l, CN ⊥ l and OL ⊥ l.∴ BM || OL || CNNow, BM | OL || CN and BC is the transversal making equal intercepts i.e., BL = LC.∴ The transversal MN will also make equal intercepts.⇒ OM = ONIn Δ LMO and Δ LNO,OM = ON    (Proved)∠LOM = ∠LON  (90°)OL = OL    (Common)∴ ΔLMO  ΔLNO  (SAS congruence criterion)⇒ LM = LN (CPCT)
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