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Let \( \mathrm{A}(4,2), \mathrm{B}(6,5) \) and \( \mathrm{C}(1,4) \) be the vertices of \( \triangle \mathrm{ABC} \).
(i) The median from A meets \( \mathrm{BC} \) at D. Find the coordinates of the point D.
(ii) Find the coordinates of the point \( \mathrm{P} \) on \( \mathrm{AD} \) such that \( \mathrm{AP}: \mathrm{PD}=2: 1 \)
(iii) Find the coordinates of points \( \mathrm{Q} \) and \( \mathrm{R} \) on medians \( \mathrm{BE} \) and \( \mathrm{CF} \) respectively such that \( \mathrm{BQ}: \mathrm{QE}=2: 1 \) and \( \mathrm{CR}: \mathrm{RF}=2: 1 \).
(iv) What do yo observe?
[Note: The point which is common to all the three medians is called the centroid and this point divides each median in the ratio 2 : 1.]
(v) If \( \mathrm{A}\left(x_{1}, y_{1}\right), \mathrm{B}\left(x_{2}, y_{2}\right) \) and \( \mathrm{C}\left(x_{3}, y_{3}\right) \) are the vertices of \( \triangle \mathrm{ABC} \). find the coordinates of the centroid of the triangle.
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