Q. 193.8( 10 Votes )

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


Answer :

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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