To determine the correct effect of the dilation with a scale factor of 4 and a center of dilation at the origin (0,0), we need to apply the dilation formula:
For a point \((x, y)\), when we apply a dilation with scale factor \(k\) from the origin, the new coordinates \((x', y')\) are given by: \[ (x', y') = (kx, ky) \]
Now, let's apply this to each of the points A, B, C, and D.
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For point A(1, 4): \[ A' = (4 \cdot 1, 4 \cdot 4) = (4, 16) \]
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For point B(2, 2): \[ B' = (4 \cdot 2, 4 \cdot 2) = (8, 8) \]
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For point C(2, 5): \[ C' = (4 \cdot 2, 4 \cdot 5) = (8, 20) \]
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For point D(3, 3): \[ D' = (4 \cdot 3, 4 \cdot 3) = (12, 12) \]
Now we can summarize the new coordinates of the points after dilation:
- A' = (4, 16)
- B' = (8, 8)
- C' = (8, 20)
- D' = (12, 12)
Looking at the options provided, the correct one is:
Option #2: After dilation, the lines are four units farther from the point of dilation. The resulting points are A′ (4,16), B′ (8,8), C′(8,20), and D′(12,12).