To find the coordinates of A′ and B′ after dilating the line segment AB with a scale factor of \( \frac{1}{3} \) centered at the origin, we need to apply the dilation formula to each endpoint.
The dilation formula for a point \( (x, y) \) with a scale factor \( k \) centered at the origin is given by: \[ (x', y') = (kx, ky) \]
Applying the dilation to point A(-6, 3):
- \( x' = \frac{1}{3} \times (-6) = -2 \)
- \( y' = \frac{1}{3} \times 3 = 1 \)
So, the new coordinates for A′ are: \[ A′(-2, 1) \]
Applying the dilation to point B(-12, 9):
- \( x' = \frac{1}{3} \times (-12) = -4 \)
- \( y' = \frac{1}{3} \times 9 = 3 \)
So, the new coordinates for B′ are: \[ B′(-4, 3) \]
Final Answer:
After dilation, the coordinates are: A′(−2, 1) and B′(−4, 3)
Thus, the correct response is: A′(−2, 1) and B′(−4, 3).