To find the points \( X' \) and \( Y' \) resulting from the dilation of the points \( X \) and \( Y \) with a scale factor of 2 and a center of dilation at the origin, we will follow these steps:
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Identify the original points \( X \) and \( Y \). Let's assume the coordinates of the original points are: \[ X = (x_1, y_1), \quad Y = (x_2, y_2) \]
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Apply the dilation formula. When we dilate a point \( (x, y) \) by a scale factor of \( k \) about the origin, the new coordinates \( (x', y') \) are given by: \[ (x', y') = (kx, ky) \] For a scale factor of \( 2 \): \[ X' = (2x_1, 2y_1), \quad Y' = (2x_2, 2y_2) \]
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Calculate the lengths of \( XY \) and \( X'Y' \). The length of a segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Length of \( XY \): \[ \text{Length of } XY = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Length of \( X'Y' \): \[ X'Y' = \sqrt{(2x_2 - 2x_1)^2 + (2y_2 - 2y_1)^2} \] Simplifying this: \[ X'Y' = \sqrt{(2(x_2 - x_1))^2 + (2(y_2 - y_1))^2} = \sqrt{4((x_2 - x_1)^2 + (y_2 - y_1)^2)} = 2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Conclusion. From our calculations:
- The length of \( XY \) is: \[ \text{Length of } XY = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- The length of \( X'Y' \) is: \[ \text{Length of } X'Y' = 2 \cdot \text{Length of } XY \]
This proves that the dilation of \( XY \) with a scale factor of \( 2 \) results in a segment \( X'Y' \) whose length is twice that of \( XY \).