To find the dilation of a shape centered at the origin by a factor of 2, you multiply each coordinate of the shape by the dilation factor (which is 2 in this case).
Given the points:
- \( a = (1, -4) \)
- \( b = (1, -1) \)
- \( c = (3, -1) \)
- \( d = (3, -4) \)
We will apply the dilation transformation to each point.
Dilation by a factor of 2:
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For point \( a = (1, -4) \): \[ a' = (2 \times 1, 2 \times -4) = (2, -8) \]
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For point \( b = (1, -1) \): \[ b' = (2 \times 1, 2 \times -1) = (2, -2) \]
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For point \( c = (3, -1) \): \[ c' = (2 \times 3, 2 \times -1) = (6, -2) \]
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For point \( d = (3, -4) \): \[ d' = (2 \times 3, 2 \times -4) = (6, -8) \]
The new points after dilation are:
- \( a' = (2, -8) \)
- \( b' = (2, -2) \)
- \( c' = (6, -2) \)
- \( d' = (6, -8) \)
So, the coordinates of the dilated shape are:
- \( a'(2, -8) \)
- \( b'(2, -2) \)
- \( c'(6, -2) \)
- \( d'(6, -8) \)
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