To determine the points A' and B' after the dilation of points A and B with a scale factor of 1/2 and center of dilation at the origin, we can use the formula for dilation:
\[ A' = (kx, ky) \]
where \( (x, y) \) are the coordinates of the original point and \( k \) is the scale factor. Here \( k = \frac{1}{2} \).
Let's assume that the points A and B are defined as:
- Point A: \( (x_1, y_1) \)
- Point B: \( (x_2, y_2) \)
To find the new points A' and B' after dilation, we will multiply each coordinate of points A and B by the scale factor \( \frac{1}{2} \):
If A is at \( (x_1, y_1) \): \[ A' = \left( \frac{1}{2}x_1, \frac{1}{2}y_1 \right) \]
If B is at \( (x_2, y_2) \): \[ B' = \left( \frac{1}{2}x_2, \frac{1}{2}y_2 \right) \]
Let's analyze the given points:
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A' is at (-1, -2) and B' is at (-3, -3)
- This implies A was at \( (-2, -4) \) and B was at \( (-6, -6) \).
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A' is at (1, 2) and B' is at (3, 3)
- This implies A was at \( (2, 4) \) and B was at \( (6, 6) \).
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A' is at (1, 5) and B' is at (3, 6)
- This implies A was at \( (2, 10) \) and B was at \( (6, 12) \).
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A' is at (4, 8) and B' is at (12, 12)
- This implies A was at \( (8, 16) \) and B was at \( (24, 24) \).
Let's prove that the length of A'B' is half of AB.
To show that the length of segment A'B' is half that of segment AB, we can use the distance formula for points A and B.
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Assuming \( A = (x_1, y_1) \) and \( B = (x_2, y_2) \):
- The length \( AB \) would be:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- The length \( A'B' \) would be:
\[ A'B' = \sqrt{\left(\frac{1}{2}x_2 - \frac{1}{2}x_1\right)^2 + \left(\frac{1}{2}y_2 - \frac{1}{2}y_1\right)^2} \] \[ A'B' = \sqrt{\left(\frac{1}{2}(x_2 - x_1)\right)^2 + \left(\frac{1}{2}(y_2 - y_1)\right)^2} \] \[ = \frac{1}{2} \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] \[ = \frac{1}{2} AB \]
Conclusion:
So, by using the above distance definitions, we can see that the length of A'B' is indeed half the length of AB when dilated by a scale factor of 1/2 from the origin. Therefore, A'B' is half the length of AB.
From the provided options, the correct choice for \( A' \) and \( B' \) that satisfies this condition is:
A' is at (1,2) and B' is at (3,3).
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