In triangle ABC ,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 = LN
Construction: Draw OL ⊥ l
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 || CN
Now, BM | OL || CN and BC is the transversal making equal intercepts i.e., BL = LC.
∴ The transversal MN will also make equal intercepts.
⇒ OM = ON
In Δ LMO and Δ LNO,
OM = ON (Proved)
∠LOM = ∠LON (90°)
OL = OL (Common)
∴ ΔLMO ΔLNO (SAS congruence criterion)
⇒ LM = LN (CPCT)