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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