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In a ΔPQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP

respectively. Prove that: LN = MN.

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#### Solution

Given that, in PQR, PQ QRand L,M,N are midpoints of the sides PQ, QP and RP

respectively and given to prove that LN MN

Here we can observe that PQR is and isosceles triangle

⇒PQ =QR and ∠QPR =∠QRP ……..(1)

And also, L and M are midpoints of PQ and QR respectively

⇒ `PL=LQ=(PQ)/2,QM=MR=(QR)/2`

And also, PQ=QR

⇒ `PL=LQ=QM=MR=(PQ)/2=(QR)/2` .............(2)

Now, consider ΔLPN and ,Δ MRN

LP= MR [From – (2)]

∠LPN =∠MRN [From – (1)]

∵∠QPR and ∠LPN and ∠ QRP and ∠MRN are same

PN= NR [∵N is midpoint of PR]

So, by SAS congruence criterion, we have LPN≅ MRN

⇒LN =MN

[ ∵Corresponding parts of congruent triangles are equal]

Concept: Congruence of Triangles

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