MandN are the mid-points of the sides QR and PQ respectively of a A
PQR, right-angled at Q. Prove that:
(i) PMⓇ + RN2 = 5 MN?
(ii) 4 PM2 = 4 PQ² + QR2
(iii) 4 RN2 = PQ2 + 4 QR2
(iv) 4 (PM? + RN2) = 5 PR2
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Answer:
Construct a line to joining point N to M
InΔPQM
PQ
2
+QM
2
=PM
2
....(1)
InΔNQR
NQ
2
+QR
2
=NR
2
...(2)
andInΔNQM
NQ
2
+QM
2
=NM
2
....(3)
Addingeq(1)and(2)
PM
2
+NR
2
=(PQ
2
+QR
2
)+(QM
2
+QN
2
)
PM
2
+NR
2
=(2QN)
2
+(2QM)
2
+NM
2
[∵MandNismidpointPQandQRrespectively]
PM
2
+NR
2
=4(QN
2
+4QM
2
)+NM
2
fromequation(3)
PM
2
+NR
2
=4NM
2
+NM
2
=5NM
2
oved
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