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. - Mathematics

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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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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 
[ ∵Corresponding parts of congruent triangles are equal] 

Concept: Congruence of Triangles
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RD Sharma Mathematics for Class 9
Chapter 12 Congruent Triangles
Exercise 12.1 | Q 2 | Page 15

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