CM and RN are respectively the medians of ΔABC and ΔPQR. If ΔABC∼ΔPQR, prove that:
(i) ΔAMC∼ΔPNR
(ii) CMRN=ABPQ
(iii) ΔCMB∼ΔRNQ
[3 MARKS]
For each proof : 1 Mark
ΔABC∼ΔPQR [Given]
⇒ABPQ=BCQR=CARP......(1)
∠A=∠P,∠B=∠Q and ∠C=∠R......(2)
(i) In ΔAMC and ΔPNR
2 AM = AB and 2 PN = PQ [∵ CM and RN are medians]
⇒2AM2PN=CARP [from (1)]
⇒AMPN=CARP
Also, ∠MAC=∠NPR [From (2)]
∴ΔAMC∼ΔPNR [SAS similarity]
(ii) Since ΔAMC∼ΔPNR
CMRN=CARP
But CARP=ABPQ [From (1)]
⇒CMRN=ABPQ
(iii) Again,
ABPQ=BCQR [From (1)]
∴CMRN=BCQR
Also,
CMRN=ABPQ=2BM2QN
⇒CMRN=BMQN
CMRN=BCQR=BMQN
∴ΔCMB∼ΔRNQ [SSS similarity]
[Note : You can also prove part (iii) by following the same method as used for proving part (i)]