# In the given figure, CM and RN are respectively the medians of ΔABC and ΔPQR. If ΔABC ~ ΔPQR, prove that:(i) ΔAMC ~ ΔPNR(ii) (iii) ΔCMB ~ ΔRNQ

Given: CM is the median of ABC and RN is the median of PQR

Also, ABC ~ PQR

To Prove: (i) AMC ~PNR

CM is median of ABC

So, …(1)

Similarly, RN is the median of PQR

So, …(2)

Given ABC ~PQR

(corresponding sides of similar triangle are proportional)

(from (1) and (2))

…(3)

Also, since ABC ~PQR

A = P …(4)

(corresponding angles of similar triangles are equal)

In AMC and PNR

A = P (from (4))

(from (3))

AMC ~PNR (by SAS similarity)

Hence Proved

(ii)To Prove:

In part (i), we proved that AMC ~PNR

So,

(corresponding sides of a similar triangle are proportional)

Therefore,

Hence Proved

(iii) CMB ~RNQ

Given ABC ~PQR

(corresponding sides of similar triangle are proportional)

(from (1) and (2))

…(5)

Also, since ABC ~PQR

B = Q …(6)

(corresponding angles of similar triangles are equal)

In CMB and RNQ

B = Q (from (6))

(from (5))

CMB ~RNQ (by SAS similarity)

Hence Proved

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