To find the coordinates of point X that partitions the directed segment \( \overline{AB} \) in the ratio \( 1:2 \), we first need the coordinates of points A and B. Let:
- \( A(x_1, y_1) \)
- \( B(x_2, y_2) \)
Point X divides the segment \( \overline{AB} \) in the ratio \( 1:2 \), meaning that \( AX:XB = 1:2 \). The formula for finding the coordinates of a point that divides a segment in a given ratio is:
\[ X\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
where \( m \) and \( n \) are the parts of the ratio. In our case, we have:
- \( m = 1 \) (the part associated with A)
- \( n = 2 \) (the part associated with B)
Substituting these values gives:
\[ X\left(\frac{1 \cdot x_2 + 2 \cdot x_1}{1 + 2}, \frac{1 \cdot y_2 + 2 \cdot y_1}{1 + 2}\right) \]
This simplifies to:
\[ X\left(\frac{x_2 + 2x_1}{3}, \frac{y_2 + 2y_1}{3}\right) \]
Now, to calculate the exact coordinates of point X, we need to substitute the coordinates of points A and B into the formula.
For example, if:
- \( A(1, 2) \)
- \( B(4, 6) \)
Then, the coordinates of point X would be:
\[ X\left(\frac{4 + 2 \cdot 1}{3}, \frac{6 + 2 \cdot 2}{3}\right) = X\left(\frac{4 + 2}{3}, \frac{6 + 4}{3}\right) = X\left(\frac{6}{3}, \frac{10}{3}\right) = X(2, \frac{10}{3}) \]
Thus, the coordinates of point X are \( \left(2, \frac{10}{3}\right) \).
You can replace \( A \) and \( B \) with their actual coordinates if they're different to find the specific coordinates of point X.