Question

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

Answer 1

Given : In $\Delta PQR,PQ=QR$

L, M and N are the mid-points of sides PQ,

QR and RP respectively. Join LM, MN and LN.

To prove : $\angle PNM=\angle PLM$

Proof : In $\Delta PQR,$

$\therefore$ M and N are the mid points of sides PR and QR respectively

$\therefore MN||PQandMN=\frac{1}{2}PQ...\left(i\right)$

$\therefore MN=PL$

Similarly, we can prove that

LM = PN

Now in $\Delta NMLand\Delta LPN$

MN = PL (Proved)

LM = PN (Proved)

LN = LN (Common)

$\therefore \Delta NML\cong \Delta LPN\left(SSSaxiom\right)$

$\therefore \angle MNL=\angle PLN(c.p.c.t.)$

and $\angle MLN=\angle LNP(c.p.c.t.)$

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

$\Rightarrow \angle PNM=\angle PLM$