To find the dilated point D' for point D (3, 3) using a center of dilation at (1, 1) and a magnitude of \( \frac{1}{2} \), we can use the following formula for dilation:
\[ D' = (1 + k \cdot (x - cx), 1 + k \cdot (y - cy)) \]
Where:
- \( (cx, cy) \) is the center of dilation (1, 1)
- \( (x, y) \) is the original point (3, 3)
- \( k \) is the dilation factor, which is \( \frac{1}{2} \)
Now we can substitute our values into the formula. The coordinates of point D are \( (x, y) = (3, 3) \), and the center of dilation is \( (cx, cy) = (1, 1) \).
Calculating the new x-coordinate:
\[ x' = 1 + \frac{1}{2} \cdot (3 - 1) = 1 + \frac{1}{2} \cdot 2 = 1 + 1 = 2 \]
Calculating the new y-coordinate:
\[ y' = 1 + \frac{1}{2} \cdot (3 - 1) = 1 + \frac{1}{2} \cdot 2 = 1 + 1 = 2 \]
Therefore, the new ordered pair for point D' is:
\[ D' = (2, 2) \]
So the answer would be:
The new ordered pair for point D' would be at \( (2, 2) \).
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