Question

In ΔPQR, PQ = QR; L,M and N are the midpoints of the sides of PQ, QR and RP respectively. Prove that LN = MN.

Answer 1

Answer:

INΔPQR ,PQ=QR...GIVEN

∴∠R=∠P     [ANGLES OPPOSITE EQUAL SIDES]

⇒1/2 PQ=1/2 QR

⇒PL=MR

IN ΔMRN AND ΔLPN

PL=MR

∠R=∠P

PN = NR     [N IS THE MIDPOINT OF PR]

∴ΔMRN≅ΔLPN  [SAS]

⇒MN=LN     [C.P.C.T]

Answer 2

Answer:

PQ=QR

L,MN are midpoints of PQ,QR,PR respctively

⇒PL=PQ=QM=MR.............................(i)

and ∠QPR=∠QRP(angles opp. to equal sides)......................................(ii)

Step-by-step explanation:

in ΔLPN and Δ MNR

PL=MR(from (i))

∠LPN=∠MRN(from (ii))

PN=NR( N is midpoint)

∴ΔLPN≅ΔMNR(By SAS congruency)

⇒LN=MN(corresponding sides)

Hence Proved