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

Proof:

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)