To determine the scale factor used in the dilation of line segment \( AB \) (from points \( A(1, -3) \) to \( B(2, -4) \)) to \( A' \) (located at \( (4, -12) \)) and \( B' \) (located at \( (8, -16) \)), we need to analyze how much the coordinates of points \( A \) and \( B \) are scaled when they are transformed into \( A' \) and \( B' \).
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Find the distance from the origin to points A and A’:
- The distance from the origin to point \( A(1, -3) \) is: \[ d_A = \sqrt{(1 - 0)^2 + (-3 - 0)^2} = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
- The distance from the origin to point \( A'(4, -12) \) is: \[ d_{A'} = \sqrt{(4 - 0)^2 + (-12 - 0)^2} = \sqrt{4^2 + (-12)^2} = \sqrt{16 + 144} = \sqrt{160} = 4\sqrt{10} \]
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Calculate the scale factor for A to A':
- The scale factor \( k \) for the dilation can be found by dividing the distance of \( A' \) by the distance of \( A \): \[ k = \frac{d_{A'}}{d_A} = \frac{4\sqrt{10}}{\sqrt{10}} = 4 \]
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Repeat for points B and B’:
- The distance from the origin to point \( B(2, -4) \) is: \[ d_B = \sqrt{(2 - 0)^2 + (-4 - 0)^2} = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]
- The distance from the origin to point \( B'(8, -16) \) is: \[ d_{B'} = \sqrt{(8 - 0)^2 + (-16 - 0)^2} = \sqrt{8^2 + (-16)^2} = \sqrt{64 + 256} = \sqrt{320} = 8\sqrt{5} \]
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Calculate the scale factor for B to B':
- The scale factor \( k \) for the dilation can also be calculated as: \[ k = \frac{d_{B'}}{d_B} = \frac{8\sqrt{5}}{2\sqrt{5}} = 4 \]
Since both calculations yield the same scale factor of \( 4 \), we conclude that the scale factor used in the dilation of line segment \( AB \) centered at the origin is:
4