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)

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