Question

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.

Answer 1

Construction - Draw LS perpendicular to MN

As the lines BM, LS and CN are the same perpendiculars, on line MN, they are parallel to each other.

As per the intercept theorem,  

If there are three or more parallel lines and the intercepts on a transverse or equivalent are rendered by them. And on every other transversal the related intercepts are equivalent too.

As MB and LS and NC are the three parallel lines and the two transversal lines are MN and BC, thus -

Thus, BL = LC (As L is the given midpoint of BC)

MS = SN ( Intercept Theorum )

In ΔMLS and ΔLSN

MS = SN

SL = LS ( Common)

∠LSM = ∠ LSN ( As LS ⊥ MN)

Thus, ΔMLS ≅ ΔLSN ( By SAS congruency)

Also, LM = LN ( By CPCT)

Answer 2

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