Solution:-
From the question it is given that, AB || CR and LM || QR
(i) We have to prove that, BM/MC = AL/LQ
Consider the ∆ARQ
LM || QR … [from the question]
So, AM/MR = AL/LQ … [equation (i)]
Now, consider the ∆AMB and ∆MCR
∠AMB = ∠CMR … [because vertically opposite angles are equal]
∠MBA = ∠MCR … [because alternate angles are equal]
Therefore, AM/MR = BM/MC … [equation (ii)]
From equation (i) and equation (ii) we get,
BM/MR = AL/LQ
(ii) Given, BM : MC = 1 : 2
AM/MR = BM/MC
AM/MR = ½ … [equation (iii)]
LM || QR … [given from equation]
AM/MR = LM/QR … [equation (iv)]
AR/AM = QR/LM
(AM + MR)/AM = QR/LM
1 + MR/AM = QR/LM
1 + (2/1) = QR/LM
3/1 = QR/LM
LM/QR = 1/3
Therefore, the ratio of LM: QR is 1: 3.