Question

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

Answer 1

In $△ABC$, BM and CN are perpendiculars on any line passing through A.

Also, $BL=LC$

We need to prove that $ML=NL$

From point L let us draw $LP\perp AN$

It is given that $BM\perp AN$, $LP\perp AN$, and $CN\perp AN$

Therefore,

$BM\parallel LP\parallel CN$

Since, L is the midpoints of BC.

Therefore intercepts made by these parallel lines on MN will also be equal

[If a transversal makes equal intercepts on three or more parallel line, then any other transversal intersecting them will also make equal intercepts.]

Thus,

$MP=NP$

Now in $△LMN,$

$MP=NP$

And $LP\perp AN$. Thus, perpendicular bisects the opposite sides.

Therefore, $△LMN$ is isosceles.

Hence $ML=NL$

Hence proved.