To find the coordinates of point \( X \) that partitions the directed segment \( AB \) in the ratio \( 1:2 \), we can use the section formula.
Given two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the coordinates of a point \( X \) that divides the segment \( AB \) in the ratio \( m:n \) are given by:
\[ X \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
In this case, we have:
- \( A(-4, -3) \) so \( x_1 = -4 \) and \( y_1 = -3 \)
- \( B(1, 2) \) so \( x_2 = 1 \) and \( y_2 = 2 \)
- The ratio \( m:n = 1:2 \) means \( m = 1 \) and \( n = 2 \)
Now, we can substitute these values into the section formula:
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Calculate the \( x \)-coordinate: \[ X_x = \frac{1 \cdot 1 + 2 \cdot (-4)}{1 + 2} = \frac{1 - 8}{3} = \frac{-7}{3} \]
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Calculate the \( y \)-coordinate: \[ X_y = \frac{1 \cdot 2 + 2 \cdot (-3)}{1 + 2} = \frac{2 - 6}{3} = \frac{-4}{3} \]
Thus, the coordinates of point \( X \) are:
\[ X \left( \frac{-7}{3}, \frac{-4}{3} \right) \]
So, the final answer is \(\left( \frac{-7}{3}, \frac{-4}{3} \right)\).
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