To find the dilated coordinates \( A' \) and \( B' \) of the line segment \( AB \) after applying a dilation with a scale factor of \( \frac{1}{3} \) centered at the origin, we can use the dilation formula:
\[ A' = (k \cdot x_A, k \cdot y_A) \] \[ B' = (k \cdot x_B, k \cdot y_B) \]
where \( k \) is the scale factor, and \( (x_A, y_A) \) and \( (x_B, y_B) \) are the coordinates of points \( A \) and \( B \).
Given:
- \( A(-6, 3) \)
- \( B(-12, 9) \)
- Scale factor \( k = \frac{1}{3} \)
Now, applying the formula:
For point \( A' \): \[ A' = \left(\frac{1}{3} \cdot (-6), \frac{1}{3} \cdot 3\right) = \left(-2, 1\right) \]
For point \( B' \): \[ B' = \left(\frac{1}{3} \cdot (-12), \frac{1}{3} \cdot 9\right) = \left(-4, 3\right) \]
Thus, the dilated coordinates after applying the scale factor are:
- \( A'(-2, 1) \)
- \( B'(-4, 3) \)
The original coordinates are:
- \( A(-6, 3) \)
- \( B(-12, 9) \)