A dilation from the origin involves multiplying each coordinate of the points by a common factor (scale factor). To determine which set of points represents a dilation of the original parallelogram JKLM, we will analyze each option to find if there is a consistent scale factor applied to the coordinates.
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Original Coordinates:
- J(2, 5)
- K(14, 5)
- L(10, -1)
- M(-2, -1)
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Checking Each Option:
Option A:
- J'(6 , 9), K'(18 , 9), L'(14 , 3), M'(2 , 3)
- Calculate scale factor:
- J: 2 * (scale factor) = 6 → scale factor = 3
- J: 5 * (scale factor) = 9 → scale factor = 1.8 (not consistent)
Not a dilation.
Option B:
- J'(2 , 20), K'(56 , 5), L'(40 , -1), M'(-2 , -4)
- Calculate scale factor:
- J: 2 * (scale factor) = 2 → scale factor = 1
- K: 14 * (scale factor) = 56 → scale factor = 4
- L: 10 * (scale factor) = 40 → scale factor = 4
- M: -2 * (scale factor) = -2 → scale factor = 1 (not consistent)
Not a dilation.
Option C:
- J'(8 , 5), K'(56 , 5), L'(40 , -1), M'(-8 , -1)
- Calculate scale factor:
- J: 2 * (scale factor) = 8 → scale factor = 4
- J: 5 * (scale factor) = 5 → scale factor = 1 (not consistent)
Not a dilation.
Option D:
- J'(8 , 20), K'(56 , 20), L'(40 , -4), M'(-8 , -4)
- Calculate scale factor:
- J: 2 * (scale factor) = 8 → scale factor = 4
- J: 5 * (scale factor) = 20 → scale factor = 4
- L: 10 * (scale factor) = 40 → scale factor = 4
- M: -2 * (scale factor) = -8 → scale factor = 4 (consistent)
Consistent scale factor of 4. This means Option D is a dilation of parallelogram JKLM.
Final Answer: Option D.