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

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

Given : In a $Δ A B C l$ is a straight line passing through the vertex A. $B M ⊥ l a n d C N ⊥ l . L$ is the mid point of BC
To prove : LM = LN
Construction : Draw $O L ⊥ l$
Proof :
If a transversal make equal intercepts on three or more parallel lines, then any other transversal intersecting them will also make equal intercepts.
$B M ⊥ l, C N ⊥ l a n d O L ⊥ l .$
$∴ B M | | O L | | C N$
Now, BM |OL|| CN and BC is the transversal making equal intercepts i.e., BL = LC.
$∴$ The transversal MN will also make equal intercepts.
$\Rightarrow O M = O N$
$I n Δ L M O a n d Δ L N O,$
$O M = O N$
$∠ L O M = ∠ L O N$ (OL is perpendicular to BC)
OL = OL (Common line)
$∴ Δ L M O ≅ Δ L N O$ (By SAS congruence criterion)
$∴ L M = L N (B y C P C T)$

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