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Question
M andN are the mid-points of the sides QR and PQ respectively of a PQR, right-angled at Q.
Prove that:
(i) PM2 + RN2 = 5 MN2
(ii) 4 PM2 = 4 PQ2 + QR2
(iii) 4 RN2 = PQ2 + 4 QR2(iv) 4 (PM2 + RN2) = 5 PR2
Solution
We draw, PM, MN, NR
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Since M and N are the mid-points of the sides QR and PQ respectively, therefore, PN = NQ, QM = RM
(i) First, we consider the ΔPQM, and applying Pythagoras theorem we get,
PM2 = PQ2 + MQ2
= ( PN + NQ )2 + MQ2
= PN + NQ2 + 2PN . NQ + MQ2
= MN2+ PN2 + 2PN.NQ ...[From, ΔMNQ, MN2 = NQ2 + MQ2] ......(i)
Now, we consider the ΔRNQ, and applying Pythagoras theorem we get,
RN2 = NQ2 + RQ2
= NQ2 + ( QM + RM )2
= NQ2 + QM2 + RM2 + 2QM .RM
= MN2 + RM2 + 2QM . RM .......(ii)
Adding (i) and (ii) we get,
PM2 + RN2 = MN2 + PN2 + 2PN.NQ + MN2 + RM2 + 2QM. RM
PM2 + RN2 = 2MN2 + PN2 + RM2 + 2PN.NQ + 2QM.RM
PM2 + RN2 = 2MN2 + NQ2 + QM2 + 2(QN2 ) + 2(QM2 )
PM2 + RN2 = 2MN2 + MN2 + 2MN2
PM2 + RN2 = 5MN2
Hence Proved.
(ii) We consider the ΔPQM, and applying Pythagoras theorem we get,
PM2 = PQ2 + MQ2
4PM2 = 4PQ2 + 4MQ2 ...[ Multiply both sides by 4]
4PM2 = 4PQ2 + 4.`(1/2 "QR")^2` ...[ MQ = `1/2` QR ]
4PM2 = 4PQ2 + 4PQ + 4 . `1/4` QR2
4PM2 = 4PQ2 + QR2
Hence Proved.
(iii) We consider the ΔRQN, and applying Pythagoras theorem we get,
RN2 = NQ2 + RQ2
4RN2 = 4NQ2 + 4QR2 ...[ Multiplying both sides by 4]
4RN2 = 4QR2 + 4 .(1/2 PQ)2 ...[ NQ = `1/2` PQ ]
4RN2 = 4QR2 + 4 .`1/4` PQ2
4RN2 = PQ2 + 4QR2
Hence Proved.
(iv) First, we consider the ΔPQM, and applying Pythagoras theorem we get,
PM2 = PQ2 + MQ2
= ( PN + NQ )2 + MQ2
= PN2 + NQ2 + 2PN.NQ + MQ2
= MN2 + PN2 + 2PN.NQ ...[ From, ΔMNQ, = MN2 = NQ2 + MQ2 ] ......(i)
Now, we consider the ΔRNQ, and applying Pythagoras theorem we get,
RN2 + NQ2 + RQ2
= NQ2 + ( QM + RM )2
= NQ2 + QM2 + RM2 + 2QM .RM
= MN2 + RM2 + 2QM . RM .......(ii)
Adding (i) and (ii) we get,
PM2 + RN2 = MN2 + PN2 + 2PN . NQ + MN2 + RM2 + 2QM. RM
PM2 + RN2 = 2MN2 + PN2 + RM2 + 2PN . NQ + 2QM . RM
PM2 + RN2 = 2MN2 + NQ2 + QM2 + 2(QN2 ) + 2(QM2 )
PM2 + RN2 = 2MN2 + MN2 + 2MN2
PM 2 + RN2 = 5MN2
4( PM2 + RN2 ) = 4.5. (NQ2 + MQ2)
4( PM2 + RN2 ) = 4.5. `[ ( 1/2 "PQ" )^2 + ( 1/2 "RQ" )^2 ] ....[ ∵ "NQ" = 1/2 "PQ" , "MQ" = 1/2 "QR" ]`
4 ( PM2 + RN2 ) = 5PR2
Hence Proved.
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