In the figure, R is the mid-point of AB, P is the mid-point of AR and L is the mid-point of AP. If RS, PQ and LM are parallel to each other, then the length of BC is :

1. 3 LM

2. 4 LM

3. 6 LM

4. 8 LM

By mid-point theorem,

The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it.

By converse of mid-point theorem,

A line drawn through the midpoint of one side of a triangle, and parallel to another side, will bisect the third side.

In △APQ,

Given,

AL = LP and LM || PQ

∴ M is mid-point of AQ (By converse of mid-point theorem)

⇒ L and M are midpoints of AP and AQ respectively.

∴ LM = $\dfrac{1}{2}$ PQ

⇒ PQ = 2 LM ……(1)

In △ARS,

Given,

AP = RP and PQ || RS

∴ Q is mid-point of AS (By converse of mid-point theorem)

⇒ P and Q are midpoints of AR and AS respectively.

∴ PQ = $\dfrac{1}{2}$ RS ………(2)

From equation (1) and (2), we get :

⇒ 2 LM = $\dfrac{1}{2}$ RS

⇒ RS = 4 LM ….(3)

In △ABC,

Given,

AR = BR

⇒ RS || BC and S is the mid-point of AC. (By converse of mid-point theorem)

∴ RS = $\dfrac{1}{2}$ BC (By mid-point theorem)

Substituting value of RS in equation (3), we get:

⇒ $\dfrac{1}{2}$ BC = 4 LM

⇒ BC = 8 LM.

Hence, option 4 is the correct option.