To find the coordinates of points Aā² and Bā² after dilation of the line segment AB with a scale factor of \(\frac{1}{3}\) centered at the origin, you can use the formula for dilation, which is given by:
\[ (x', y') = (kx, ky) \]
where \( (x, y) \) are the coordinates of the original point, \( (x', y') \) are the coordinates of the dilated point, and \( k \) is the scale factor.
Given:
- A(-6, 3)
- B(-12, 9)
- Scale factor \( k = \frac{1}{3} \)
Step 1: Calculate Aā²
For point A(-6, 3): \[ A' = \left( \frac{1}{3} \times -6, \frac{1}{3} \times 3 \right) = \left( -2, 1 \right) \]
Step 2: Calculate Bā²
For point B(-12, 9): \[ B' = \left( \frac{1}{3} \times -12, \frac{1}{3} \times 9 \right) = \left( -4, 3 \right) \]
Conclusion:
The coordinates after dilation are:
- Aā² = (-2, 1)
- Bā² = (-4, 3)