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In figure Cm and RN are respectively th...

In figure Cm and RN are respectively the medians of `DeltaA B C`and `DeltaP Q R`. If `DeltaA B C ~DeltaP Q R`, prove that:
(i) `DeltaA M C ~DeltaP N R`
(ii) `(C M)/(R N)=(A B)/(P Q)`
(ii) `DeltaC M B ~DeltaR N Q`

Text Solution

Verified by Experts

(i) `ΔABC ~ΔPQR` (Given)
So,`(AB)/(PQ)=(BC)/(QR)=(CA)/(RP)` (1)
and
`/_A =/_P, /_B =/_Q and /_C =/_R` (2)
But `AB = 2 AM` and `PQ = 2 PN`
(As CM and RN are medians)
So, from (1),
`(2AM)/(2PN)=(CA)/(RP)`
i.e.,`(AM)/(PN)=(CA)/(RP)` (3)
Also, `/_MAC =/_NPR` [From (2)] (4)
So, from (3) and (4),
`triangleAMC ~trianglePNR` (SAS similarity) (5)
(ii) From (5),`(CM)/(RN)=(CA)/(RP)` (6)
But `(CA)/(RP)=(AB)/(PQ)` [From (1)] (7)
Therefore, `(CM)/(RN)=(AB)/(PQ)` [From (6) and (7)] (8)

(iii) Again, `(AB)/(PQ)=(BC)/(QR)` [From (1)]
Therefore, `(CM)/(RN)=(BC)/(QR)` [From (8)] (9)
Also, `(CM)/(RN)=(AB)/(PQ)=(2BM)/(2QN)`
i.e.,`(CM)/(RN)=(BM)/(QN)` (10)
i.e., `(CM)/(RN)=(BC)/(QR)=(BM)/ (QN)` [From (9) and (10)]
Therefore,`triangleCMB ~triangleRNQ "(SSS similarity)"`
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