To find the coordinates \( A' \) and \( B' \) after a dilation of the line segment \( AB \) with a scale factor of \( 13 \) centered at the origin, we can apply the dilation formula:
\[ A' = (x_A \cdot k, y_A \cdot k) \quad \text{and} \quad B' = (x_B \cdot k, y_B \cdot k) \]
where \( (x_A, y_A) \) and \( (x_B, y_B) \) are the original coordinates of points \( A \) and \( B \), and \( k \) is the scale factor.
For point \( A(-6, 3) \):
- \( x_A = -6 \)
- \( y_A = 3 \)
For point \( B(-12, 9) \):
- \( x_B = -12 \)
- \( y_B = 9 \)
Given \( k = 13 \):
Now, let's calculate the new coordinates:
-
For point \( A' \): \[ A' = (-6 \cdot 13, 3 \cdot 13) = (-78, 39) \]
-
For point \( B' \): \[ B' = (-12 \cdot 13, 9 \cdot 13) = (-156, 117) \]
Thus, the coordinates after dilation are:
- \( A'(-78, 39) \)
- \( B'(-156, 117) \)