To find the new coordinates of the vertices of a parallelogram after dilation with respect to the origin by a factor of 3, you need to multiply each coordinate of the vertices by 3.
Let’s say the original coordinates of the vertices are:
- E(x₁, y₁)
- F(x₂, y₂)
- G(x₃, y₃)
- H(x₄, y₄)
After dilation, the new coordinates will be:
- E′(3x₁, 3y₁)
- F′(3x₂, 3y₂)
- G′(3x₃, 3y₃)
- H′(3x₄, 3y₄)
If we assume the original coordinates of points E, F, G, and H are known, we can perform the multiplication.
However, since you've presented only the transformed coordinates, we will verify them by checking if they can be derived from any original coordinates by multiplication by 3.
-
E′(−3,15) , F′(6,24) , G′(12,12), and H′(3, 3)
- Original E would be: (-1, 5)
- Original F would be: (2, 8)
- Original G would be: (4, 4)
- Original H would be: (1, 1)
- All original coordinates are correct since they multiply by 3 to yield the new coordinates.
-
E′(−13,53) , F′(23,83) , G′(43,43) , and H′(13,13)
- Original E would be: (-13/3, 53/3) which is not an integer.
- Original F would be: (23/3, 83/3) which is not an integer.
- This set doesn't yield integer coordinates.
-
E′(−4,2) , F′(−1,5) , G′(1,1) and H′(−2,−2)
- Original E would be: (-4/3, 2/3) which is not an integer.
- Original F would be: (-1/3, 5/3) which is not an integer.
- This set doesn't yield integer coordinates.
-
E′(−13,53) , F′(23,83) , G′(43,43) and H′(13,13)
- Same as option 2.
The only set of coordinates that logically follows the dilation process is the first option: E′(−3,15) , F′(6,24) , G′(12,12), and H′(3, 3).
Thus, the correct answer is: E′(−3,15) , F′(6,24) , G′(12,12), and H′(3, 3).
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