Question

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

Answer 1

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)

∴ Δ LMO ~= Δ LNO  (SAS congruence criterion)

⇒ LM = LN  (Corresponding part of congruent triangle)

Answer 2

Hope it must help u in the given fig