Question

5. In a APQR, if PQ = QR and L, M and N are the mid-
points of the sides PQ, QR and RP respectively. Prove
that LN = MN.​

Answer 1

Answer:

Using mid-point theorem,

we have

MN∥PQ and MN=

2

1

PQ⇒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

ΔNML≅ΔLPN

⇒∠MNL=∠PLN and ∠MLN=∠LNP

⇒∠MNL=∠LNP=∠PLM=∠MLN

⇒∠PNM=∠PLM

∴LN=MN

Answer 2

Answer: Given that in Δ PQR, PQ=QR and L, M and N are the mid-point of the sides PQ, QR and RP respectively.

We need to prove that LN = MN

Since, it is given that  PQ=QR

By the property of triangle, Angles opposite to the equal sides of a triangle are equal.

∠R=∠P

Thus, we have,

1/2PQ=1/2QR

PL=MR

Now, we shall consider the triangles Δ MRN and Δ LPN

PL=MR

∠R=∠P

Since, N is the midpoint of PR, we have,

PN=NR

By SAS property, we get,

ΔMRN ≅ΔLPN

Then by Corresponding Parts of Congruent Triangles theorem, we have,

MN=LN