To find the images of points A and B after the dilation around the origin with a scale factor of \( \frac{1}{2} \), we will use the following formula for dilation:
\[ (x', y') = (k \cdot x, k \cdot y) \]
where \( (x, y) \) are the original coordinates, \( (x', y') \) are the new coordinates after dilation, and \( k \) is the scale factor.
Let’s check the points provided.
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Option A: A' is at (4,8) and B' is at (12,12)
If \( A = (4, 8) \) and \( B = (12, 12) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot 4, \frac{1}{2} \cdot 8\right) = (2, 4) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot 12, \frac{1}{2} \cdot 12\right) = (6, 6) \)
Hence, A' should be (2,4) and B' should be (6,6). So, this option is incorrect.
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Option B: A' is at (1,5) and B' is at (3,3)
If \( A = (1, 5) \) and \( B = (3, 3) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot 1, \frac{1}{2} \cdot 5\right) = \left(\frac{1}{2}, \frac{5}{2}\right) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot 3, \frac{1}{2} \cdot 3\right) = \left(\frac{3}{2}, \frac{3}{2}\right) \)
Hence, A' = (1/2, 2.5) and B' = (1.5, 1.5). This option is incorrect.
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Option C: A' is at (1,2) and B' is at (3,3)
If \( A = (1, 2) \) and \( B = (3, 3) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot 1, \frac{1}{2} \cdot 2\right) = \left(\frac{1}{2}, 1\right) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot 3, \frac{1}{2} \cdot 3\right) = \left(\frac{3}{2}, \frac{3}{2}\right) \)
Hence, A' = (1/2, 1) and B' = (1.5, 1.5). This option is incorrect.
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Option D: A' is at (-1,-2) and B' is at (-3, -3)
If \( A = (-1, -2) \) and \( B = (-3, -3) \):
- Dilation of A: \( \left(\frac{1}{2} \cdot (-1), \frac{1}{2} \cdot (-2)\right) = \left(-\frac{1}{2}, -1\right) \)
- Dilation of B: \( \left(\frac{1}{2} \cdot (-3), \frac{1}{2} \cdot (-3)\right) = \left(-\frac{3}{2}, -\frac{3}{2}\right) \)
Hence, A' = (-1/2, -1) and B' = (-1.5, -1.5). This option is incorrect.
None of the options given correspond to the correct dilation of line AB with a scale factor of \( 1/2 \).
Nonetheless, to prove that the length of line A'B' is half the length of line AB, we need the coordinates of A and B originally given, for which no options provide that.
To prove this with existing coordinates:
Let \( A = (x_A, y_A) \) and \( B = (x_B, y_B) \). The distance AB is given by: \[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \] The length of A'B' after dilation of A and B: \[ A'B' = \sqrt{(kx_B - kx_A)^2 + (ky_B - ky_A)^2} = k \cdot \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \frac{1}{2} \cdot AB. \]
Thus, the length of A'B' is indeed half the length of AB due to the properties of dilation.