Question

CM and RN are respectively the medians of triangle ABC and triangle PQR. if ABC ~PQR, prove that : AMC~ PNR ​CM / RN =AB/PQ CMB ~ RNQ

Answer 1

from the above the diagrams CM and RN are the medians for the triangles.

we know that ΔABC≈ ΔPQR

AB/PQ = BC/QR = AC/PR ------------------------(1)
fromΔAMC and ΔPNR
∠A = ∠P
∠M = ∠N = 90
AC = PR
by ASA similarity 
ΔAMC ≈ΔPNR         ----PROVED
from this 
AC/PR = CM/RN = AM/PN
FROM ------(1)
AB/PQ = BC/QR = AC/PR
from that
CM/RN = AB/PQ     -----PROVED
from ΔBMC and ΔQNR
∠B = ∠Q
∠M = ∠N = 90
BC = RQ
by ASA similarity 
ΔBMC≈ΔQNR         ------PROVED

Answer 2

Answer:

Step-by-step explanation:

(1) ABC~PQR

AB/PQ=BC/QR=CA/RP. (1)

angleA=angleP, angleB=angleQ, angleC=angleR. (2)

AB=2AM and PQ=2PN.

(as CM and RN are medians)

From (1)

2AM/2PN=CA/RP

AM/PN=CA/RP. (3)

angleMAC=angleNPR. [FROM (2)] (4)

FROM (3), (4)

TRIANGLEamc~triangle PNR. (SAS)(5)

(2) FROM (5)

CM/RN=CA/RP

CA/RP=CA/RP