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  4. P, Q, and R are on A B, B C, and A C of the equilateral triangle A B C, respectively. A P: P B=C Q: Q B=1: 2. G is the centroid of the triangle P Q B and R is the mid-point of A C. Find B G: G R.

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Q. $P, Q$, and $R$ are on $A B, B C$, and $A C$ of the equilateral triangle $A B C$, respectively. $A P: P B=C Q: Q B=1$ : 2. $G$ is the centroid of the triangle $P Q B$ and $R$ is the mid-point of $A C$. Find $B G: G R$.

Geometry

Solution:

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Let $A B=B C=A C=3 x$
$\therefore B P=B Q=P Q=2 x$
$(\because A P: B P=C Q: B Q=1: 2)$
As $\triangle B P Q$ and $\triangle B A C$ are equilateral triangle, the centroid of $\triangle B P Q$ lies on $B R$ (where $B R$ is median drawn on to $A C$ )
We know that centroid divides the median in the ratio $2: 1$.
$\therefore B G=\frac{2}{3} \frac{[\sqrt{3}(2 x)]}{2}=\frac{2 \sqrt{3} x}{3}$
But $B R=\frac{\sqrt{3}(3 x)}{2}=\frac{3 \sqrt{3} x}{2}$
Now $G R=B R-B G$
$=\frac{3 \sqrt{3} x}{2}-\frac{2 \sqrt{3} x}{3}=\frac{5 \sqrt{3} x}{6}$
Now $B G: G R=\frac{2 \sqrt{3} x}{3}: \frac{5 \sqrt{3} x}{6}=4: 5$.