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Question
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]
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