To perform a dilation of a point using a center of dilation, we can use the following formula for the coordinates of the dilated point:
\[ (x', y') = \left( x_c + k(x - x_c), y_c + k(y - y_c) \right) \]
where:
- \((x', y')\) are the coordinates of the dilated point,
- \((x, y)\) are the coordinates of the original point,
- \((x_c, y_c)\) are the coordinates of the center of dilation,
- \(k\) is the magnitude of the dilation.
In this case, your coordinates for point \(D\) are \(D(3, 3)\), the center of dilation is \((1, 1)\), and the magnitude of dilation is \(k = \frac{1}{2}\).
Now we can substitute the values into the dilation formula:
- \(x_c = 1\), \(y_c = 1\)
- \(x = 3\), \(y = 3\)
- \(k = \frac{1}{2}\)
Calculating the new coordinates: \[ x' = 1 + \frac{1}{2}(3 - 1) = 1 + \frac{1}{2} \cdot 2 = 1 + 1 = 2 \]
\[ y' = 1 + \frac{1}{2}(3 - 1) = 1 + \frac{1}{2} \cdot 2 = 1 + 1 = 2 \]
Thus, the ordered pair for the point \(D\) after the dilation is \(D'(2, 2)\).