To find the coordinates of point P that partitions the directed line segment from S(-2, 4) to T(7, -2) in the ratio 1:2, follow these steps:
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Determine the ratio: The ratio of 1:2 means that point P divides the segment into 1 part from S to P and 2 parts from P to T.
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Find the coordinates using the section formula: The section formula for finding a point P that divides a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) is given by: \[ P\left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right) \] Here, \( S(-2, 4) = (x_1, y_1) \) and \( T(7, -2) = (x_2, y_2) \), with \( m = 1 \) and \( n = 2 \).
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Plug in the values: \[ P_x = \frac{1 \cdot 7 + 2 \cdot (-2)}{1 + 2} = \frac{7 - 4}{3} = \frac{3}{3} = 1 \] \[ P_y = \frac{1 \cdot (-2) + 2 \cdot 4}{1 + 2} = \frac{-2 + 8}{3} = \frac{6}{3} = 2 \]
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Write the coordinates of point P: The coordinates of point P are \( (1, 2) \).
Thus, point P that partitions the segment from S to T in the ratio 1:2 is at coordinates \( (1, 2) \).