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प्रश्न
Find the centroid of a triangle, mid-points of whose sides are (1, 2, –3), (3, 0, 1) and (–1, 1, –4).
Notes
\[\text{ Let A }\left( x_1 , y_1 , z_1 \right), B\left( x_2 , y_2 , z_2 \right) \text{ and } C\left( x_3 , y_3 , z_3 \right) \text{ be the vertices of the given triangle }, \]
\[\text{ and let } D\left( 1, 2, - 3 \right) , E\left( 3, 0, 1 \right) \text{ and } F\left( - 1, 1, - 4 \right) \text{ be the mid points of the sides BC, CA and AB } respectively . \]
\[\text{ D is the mid point of BC }\]
\[ \therefore \frac{x_2 + x_3}{2} = 1, \frac{y_2 + y_3}{2} = 2 \text{ and } \frac{z_2 + z_3}{2} = - 3\]
\[ \Rightarrow x_2 + x_3 = 2, y_2 + y_3 = 4 \text{ and } z_2 + z_3 = - 6 . . . \left( 1 \right)\]
\[\text{ E is the mid point of CA }\]
\[ \therefore \frac{x_1 + x_3}{2} = 3, \frac{y_1 + y_3}{2} = 0 \text{ and } \frac{z_1 + z_3}{2} = 1\]
\[ \Rightarrow x_1 + x_3 = 6, y_1 + y_3 = 0 \text{ and } z_1 + z_3 = 2 . . . \left( 2 \right)\]
\[\text{ F is the mid point of AB }\]
\[ \therefore \frac{x_1 + x_2}{2} = - 1, \frac{y_1 + y_2}{2} = 1 \text{ and } \frac{z_1 + z_2}{2} = - 4\]
\[ \Rightarrow x_1 + x_2 = - 2, y_1 + y_2 = 2 \text{ and } z_1 + z_2 = - 8 . . . \left( 2 \right)\]
\[\text{ Adding first three equations we get }, \]
\[2\left( x_1 + x_2 + x_3 \right) = 6, 2\left( y_1 + y_2 + y_3 \right) = 6 \text{ and } 2\left( z_1 + z_2 + z_3 \right) = - 12\]
\[ \Rightarrow x_1 + x_2 + x_3 = 3, y_1 + y_2 + y_3 = 3 \text{ and } z_1 + z_2 + z_3 = - 6\]
\[\text{ The coordinate of the centroid is given by } \]
\[\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right)\]
\[ \Rightarrow \left( 1, 1, - 2 \right)\]
\[\]
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