Mathematics

# In a $\Delta PQR,$ If PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively. Prove that $LN = MN.$

##### SOLUTION

Using mid-point theorem,

we have

$MN \parallel PQ$ and $MN = \dfrac{1}{2}PQ\Rightarrow MN = PL$

Similarly, we have

$LM = PN$

In triangles $NML$ and $LPN,$ we have

$MN = PL$

$LM = PN$

and, $LN = NL$

So, by $SSS$ congruence criterion, we obtain

$\Delta NML \cong \Delta LPN$

$\Rightarrow \angle MNL = \angle PLN$ and $\angle MLN = \angle LNP$

$\Rightarrow \angle MNL = \angle LNP = \angle PLM = \angle MLN$

$\Rightarrow \angle PNM = \angle PLM$

$\therefore LN=MN$

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Subjective Medium Published on 09th 09, 2020
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