Question

In a ΔABC, BM and CN are perpendiculars from B and C respectively on any line passing through A. If L is the mid-point of BC, prove that ML=NL.

Answer 1

In a ΔABC, BM and CN are perpendiculars from B and C respectively on any line passing through A. If L is the mid-point of BC,

Hence it is proved that ML=NL.

Given,

L is the mid-point of BC

⇒ BL = CL    

From figure it's clear that,

BM ║ OL ║ CN

BL = LC (given - L is the mid-point of BC)

As MN is traversal,

MO = ON (O is the mid-point of MN).... (1)

In Δ LMO and Δ LNO

OM = ON (from (1))

∠ LOM = ∠ LON = 90°

OL = OL (common side)

Therefore, from SAS congruence criteria, we have,

Δ LMO ≅ Δ LNO

As we know that, the congruent sides of the congruent triangles are same.

∴ ML = LN

Hence the proof.

Answer 2

see this attachment.................