In ∆ABC, P and Q are the middle points of the sides AB and AC, respectively. R is a point on the segment PQ such that PR: RQ = 1:5. If PR= 6 cm, then BC = ____.

The correct answer is 72 cm.

Step-by-Step Solution:

1. Understand the given information:

In $△\text{ABC}$, $\text{P}$ and $\text{Q}$ are the midpoints of sides $\text{AB}$ and $\text{AC}$, respectively.
$\text{R}$ is a point on the segment $\text{PQ}$ such that the ratio $\text{PR}:\text{RQ}=1:5$.
We are given that $\text{PR}=6\text{cm}$.

2. Calculate the total length of segment PQ:

Since $\text{PR}:\text{RQ}=1:5$, let $\text{PR}=x$ and $\text{RQ}=5x$.
Given $\text{PR}=6\text{cm}$, we have $x=6\text{cm}$.
Therefore, $\text{RQ}=5\times 6=30\text{cm}$.

The total length of $\text{PQ}$ is:

$\text{PQ}=\text{PR}+\text{RQ}$

$\text{PQ}=6\text{cm}+30\text{cm}=36\text{cm}$

3. Apply the Midpoint Theorem:

According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is equal to half of its length.

Thus, for $△\text{ABC}$:

$\text{PQ}=\frac{1}{2}\text{BC}$

$\text{BC}=2\times \text{PQ}$

4. Calculate the length of BC:

$\text{BC}=2\times 36\text{cm}=72\text{cm}$

Hence, the length of side $\text{BC}$ is 72 cm.