To prove that the length of segment A'B' is half the length of segment AB after a dilation with a center of dilation at the origin and a scale factor of \( \frac{1}{2} \), we need to verify the points for A' and B' and apply the dilation transformation correctly.
Given points before dilation:
Let’s denote points A and B as follows:
- Let \( A(x_1, y_1) \)
- Let \( B(x_2, y_2) \)
Dilation formula:
When performing a dilation about the origin with a scale factor \( k \), the coordinates of the dilated points are given by:
- \( A' = (k \cdot x_1, k \cdot y_1) \)
- \( B' = (k \cdot x_2, k \cdot y_2) \)
In this case, the scale factor \( k \) is \( \frac{1}{2} \). Therefore, we have:
- \( A' = \left(\frac{1}{2} x_1, \frac{1}{2} y_1\right) \)
- \( B' = \left(\frac{1}{2} x_2, \frac{1}{2} y_2\right) \)
Testing the options provided:
To find which points correspond to the points after dilation, we can check the transformation of each pair of points given:
-
Option 1: \( A'(-1, -2) \) and \( B'(-3, -3) \)
- This indicates the original points were at \( A(2, 4) \) and \( B(6, 6) \).
- Length \( AB = \sqrt{(2 - 6)^2 + (4 - 6)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \).
- Length \( A'B' = \sqrt{(-1 - (-3))^2 + (-2 - (-3))^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \).
- Ratio \( \frac{A'B'}{AB} = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2} \).
-
Option 2: \( A'(1, 2) \) and \( B'(3, 3) \)
- Original points are \( A(2, 4) \) and \( B(6, 6) \).
- \( A'B' = \sqrt{(1 - 3)^2 + (2 - 3)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \).
- Ratio \( \frac{A'B'}{AB} = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2} \).
-
Option 3: \( A'(1, 5) \) and \( B'(3, 6) \)
- Original points must be \( A(2, 10) \) and \( B(6, 12) \).
- Length calculation will not return \( \frac{1}{2} \).
-
Option 4: \( A'(4, 8) \) and \( B'(12, 12) \)
- Original points must scale to \( A(8, 16) \) and \( B(24, 24) \).
- Length calculation will not return \( \frac{1}{2} \) as well.
Conclusion:
Based on the findings for the corresponding points and lengths:
- Options 1 and 2 both satisfy the condition, but it looks like Option 1 corresponds to a valid transformation from the dilated points, affirming that \( A'B' \) is indeed half of the length of \( AB \).
Thus, A'B' is half the length of AB after the dilation.