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 $BC$ , then$LM=LN.$

• A. True
• B. False

Answer 1

The correct option is A True

$l$ is a straight line passing through the vertex $A.$

$BM\perp l$ and $CN\perp l.$

$L$ is the mid-point of $BC$

Draw $OL\perp l$

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

$BM\perp l,$ $CN\perp l$ and $OL\perp l.$

$\therefore$ $BM\parallel OL\parallel CN$

Now, $BM\parallel OL\parallel CN$ and $BC$ is the transversal making equal intercepts.

i.e. $BL=LC.$

$\therefore$ The transversal $MN$ will also equal intercepts.

$\Rightarrow$ $OM=ON$

In $△LM$ and $△LNO,$

$\Rightarrow$ $OM=ON$

$\Rightarrow$ $\angle LOM=\angle LON$ [ $OL$ is perpendicular to $BC$ ]

$\Rightarrow$ $OL=OL$ [ Common line ]

$\therefore$ $△LMO\cong △LNO$ [ By $SAS$ congruence theorem ]

$\Rightarrow$ $LM=LN$ [ By C.P.C.T ]