Question

In a triangle ABC, BM and CN are perpendiculars from B and C respectively on any line passing through A. If L is the mid-point of BC, then prove that ML = NL.
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Answer 1

ANSWER:-

Given:

l is a straight line passing through the vertex A of ∆ABC.

BM ⊥l & CN ⊥l

L is the mid-point of BC.

To prove:

ML = NL

Construction:

Draw OL ⊥l

Proof:

If a transversal make equal intercept on three or more parallel line, they any other transversal intersecting then will also make equal intercepts.

BM⊥ l, CN ⊥l & OL ⊥l

Therefore,

BM||OL||CN

Now,

BM||OL||CN and BC is the transversal Making equal intercepts BL= LC.

Therefore,

The transversal MN will also make equal intercepts.

=) OM=ON

In ∆LMO & ∆LNO

OM= ON [proved]

angle LOM = angle LON [each 90°]

OL = OL [common]

Therefore,

∆LMO ≅ ∆LNO [SAS congruence rule]

=) ML = NL [c.p.c.t]

Proved.

Hope it helps ☺️

Answer 2

Answer:

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