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

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Solution

Given: X is a straight line passing through the vertex A of $Δ A B C . B M ⊥ l$ and $C N ⊥ X$ and L is the mid point of BC.
ML and NL are joined.
To prove : LM =LN
Construction : Draw $O L ⊥ l$
Proof : Consider two triangle $Δ B L M$ and $Δ C L N$
$∠ B M L = ∠ C N L = 90^{\circ}$
BL=LC as L is a mid point BC.
$∠ M L B = ∠ N L C$ as vertically opposite angles.
Therefore, $Δ B L M ≅ Δ C L N$ .
Hence, LM=LN as corresponding sides of congruent triangles are equal

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