Discussion
If three points P, Q, R lie on the side BC of the ΔABC such that BP = PQ = QR = RC. If G be centroid of ΔABC, then the ratio of areas of ΔPGR and ΔABC is :
Answer: D
The ratio is 1 : 6.
The points divide BC into four equal parts, so PR, which spans two of them, is half of BC. The centroid G lies one-third of the way up from BC, so its height above BC is one-third of the height \(h\) of the triangle:
\[ \begin{aligned} [\text{PGR}] &= \tfrac{1}{2} \cdot \tfrac{BC}{2} \cdot \tfrac{h}{3} \\ &= \tfrac{1}{6} \cdot \tfrac{1}{2} \cdot BC \cdot h \\ &= \tfrac{1}{6}[\text{ABC}] \end{aligned} \]
1 : 3 is the ratio for the triangle GBC, with the whole of BC as base. 1 : 2 and 1 : 4 forget either the halved base or the one-third height.
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