In the figure given below, $A B ∥ C R$ and $L M ∥ Q R$ .
(i) Prove that $B M / M C = A L / L Q$
(ii) Calculate $L M : Q R$ , given that $B M : M C = 1 : 2$ .

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Solution

From the question it is given that, $A B ∥ C R$ and $L M ∥ Q R$
(i) We have to prove that, $B M / M C = A L / L Q$
Consider the $△ A R Q$
$L M ∥ Q R$ … [from the question]
So, $A M / M R = A L / L Q$ … [equation (i)]
Now, consider the $△ A M B$ and $△ M C R$
$∠ A M B = ∠ C M R$ … [because vertically opposite angles are equal]
$∠ M B A = ∠ M C R$ … [because alternate angles are equal]
Therefore, $A M / M R = B M / M C$ … [equation (ii)]
From equation (i) and equation (ii) we get,
$B M / M R = A L / L Q$
(ii) Given, $B M : M C = 1 : 2$
$A M / M R = B M / M C$
$A M / M R = \frac{1}{2}$ … [equation (iii)]
$L M ∥ Q R$ … [given from equation]
$A M / M R = L M / Q R$ … [equation (iv)]
$A R / A M = Q R / L M$
$(A M + M R) / A M = Q R / L M$
$1 + M R / A M = Q R / L M$
$1 + (2 / 1) = Q R / L M$
$3 / 1 = Q R / L M$
$L M / Q R = 1 / 3$
Therefore, the ratio of $L M : Q R$ is $1 : 3$ .

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