To determine the scale factor used in the dilation of line segment \( AB \), we need to find the distances from the origin to both segments before and after dilation.
Original Points
- Point \( A \): \((-1, -3)\)
- Point \( B \): \((-3, -5)\)
New Points After Dilation
- Point \( A' \): \((-3, -9)\)
- Point \( B' \): \((-9, -15)\)
Distance from the Origin (0, 0)
We can use the distance formula to find the distances of points \( A \) and \( B \) from the origin, and then compare that to the distances of points \( A' \) and \( B' \).
Distance of A from the Origin \[ d_A = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
Distance of B from the Origin \[ d_B = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \]
Distance of A' from the Origin \[ d_{A'} = \sqrt{(-3)^2 + (-9)^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10} \]
Distance of B' from the Origin \[ d_{B'} = \sqrt{(-9)^2 + (-15)^2} = \sqrt{81 + 225} = \sqrt{306} = 3\sqrt{34} \]
Finding the Scale Factor
The scale factor \( k \) can be calculated using the formula: \[ k = \frac{\text{Distance of new point}}{\text{Distance of original point}} \]
For Point A: \[ k_A = \frac{d_{A'}}{d_A} = \frac{3\sqrt{10}}{\sqrt{10}} = 3 \]
For Point B: \[ k_B = \frac{d_{B'}}{d_B} = \frac{3\sqrt{34}}{\sqrt{34}} = 3 \]
Conclusion
The scale factor used in the dilation of line segment \( AB \) centered at the origin is the same for both points.
Thus, the scale factor is \( 3 \).
The answer is:
3