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.