To determine the scale factor used in the dilation of line segment AB with the center of dilation at the origin, we need the coordinates of the points A and B before and after the dilation.
Considering you have provided different points, let's break it down:
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Identify the original coordinates of points A and B. It seems like you've provided the points:
- A (original): \((-6, 8)\)
- B (original): \((4, -8)\)
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Identify the coordinates of the points after dilation (if provided). It seems like you also provided:
- A' (dilated): \((-3, 4)\)
- B' (dilated): \((2, -4)\)
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Calculate the scale factor using the formula for dilation: \[ \text{Scale Factor} = \frac{\text{Length of segment after dilation}}{\text{Length of segment before dilation}} \]
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Calculate the length of the original line segment AB and the dilated line segment A'B':
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Length of segment AB: \[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Using A \((-6, 8)\) and B \((4, -8)\): \[ AB = \sqrt{(4 - (-6))^2 + (-8 - 8)^2} = \sqrt{(4 + 6)^2 + (-8 - 8)^2} = \sqrt{10^2 + (-16)^2} \] \[ = \sqrt{100 + 256} = \sqrt{356} \]
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Length of segment A'B': Using A' \((-3, 4)\) and B' \((2, -4)\): \[ A'B' = \sqrt{(2 - (-3))^2 + (-4 - 4)^2} = \sqrt{(2 + 3)^2 + (-4 - 4)^2} = \sqrt{5^2 + (-8)^2} \] \[ = \sqrt{25 + 64} = \sqrt{89} \]
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Calculate the scale factor: \[ \text{Scale Factor} = \frac{\sqrt{89}}{\sqrt{356}} = \frac{\sqrt{89}}{2\sqrt{89}} = \frac{1}{2} \]
If you stated that the scale factor is not \( \frac{1}{2} \), it could be due to a misunderstanding regarding the points before or after dilation. Please double-check the coordinates or share any other details to clarify.