To find the coordinates of the points A′, B′, C′, and D′ after a dilation with a scale factor of 5 and a center of dilation at the origin, we can use the formula for dilation:
\[ (x', y') = (kx, ky) \]
where \( (x, y) \) are the original coordinates, \( (x', y') \) are the dilated coordinates, and \( k \) is the scale factor.
Given a scale factor of \( k = 5 \):
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For point A(1, -3): \[ A' = (5 \times 1, 5 \times -3) = (5, -15) \]
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For point B(4, -3): \[ B' = (5 \times 4, 5 \times -3) = (20, -15) \]
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For point C(4, -1): \[ C' = (5 \times 4, 5 \times -1) = (20, -5) \]
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For point D(1, -1): \[ D' = (5 \times 1, 5 \times -1) = (5, -5) \]
After performing the calculations, we find the new coordinates:
- A′(5, -15)
- B′(20, -15)
- C′(20, -5)
- D′(5, -5)
Thus, the correct answer is:
A′(5, −15), B′(20, −15), C′(20, −5), and D′(5, −5).
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