To find the endpoints of \( A' \) and \( B' \) after the dilation of points \( A(6, -12) \) and \( B(-3, 9) \) with a scale factor of \( \frac{1}{3} \) and a center of dilation at the origin \( (0, 0) \), we can use the formula for dilation from the origin, which is:
\[ (x', y') = (k \cdot x, k \cdot y) \]
where \( (x, y) \) are the original coordinates, \( (x', y') \) are the coordinates after dilation, and \( k \) is the scale factor.
For point A(6, -12):
- \( x' = \frac{1}{3} \cdot 6 = 2 \)
- \( y' = \frac{1}{3} \cdot (-12) = -4 \)
Thus, the dilated point \( A' \) is \( (2, -4) \).
For point B(-3, 9):
- \( x' = \frac{1}{3} \cdot (-3) = -1 \)
- \( y' = \frac{1}{3} \cdot 9 = 3 \)
Thus, the dilated point \( B' \) is \( (-1, 3) \).
Putting it all together, the endpoints after dilation are:
- \( A' (2, -4) \)
- \( B' (-1, 3) \)