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Question

CM and RN are respectively the medians of ΔABC and ΔPQR. If ΔABCΔPQR, prove that:

(i) ΔAMCΔPNR

(ii) CMRN=ABPQ

(iii) ΔCMBΔRNQ
[3 MARKS]

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Solution

For each proof : 1 Mark


ΔABCΔPQR [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)]

ΔAMCΔPNR [SAS similarity]

(ii) Since ΔAMCΔPNR

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

ΔCMBΔRNQ [SSS similarity]

[Note : You can also prove part (iii) by following the same method as used for proving part (i)]


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