# BM and CN are perpendiculars to a line passing through the vertex A of a triangle ABC. If L is mid point of BC prove that LM=LN

1
by rekhabatra

2016-03-01T14:50:43+05:30

you can see the diagram given below.

Sol:

Given: In a ΔABC

l is a straight line passing through the vertex A . 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 lines, then any other transversal intersecting them 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

∠LOM = ∠LON  (OL is perpendicular to BC)

OL = OL    (Common line )

∴ ΔLMO ≅ ΔLNO  (By SAS congruence criterion)

∴ LM = LN ( By CPCT)

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