To find the coordinates of point \( D' \) after dilating point \( D(9, 6) \) with a scale factor of \( \frac{2}{3} \) and center at the origin \( (0,0) \), we can use the formula for dilation:
\[ D' = \left( k \cdot x, k \cdot y \right) \]
where \( k \) is the scale factor, \( x \) is the x-coordinate of the original point, and \( y \) is the y-coordinate of the original point.
Given:
- \( k = \frac{2}{3} \)
- \( D(9, 6) \)
We calculate the coordinates of \( D' \):
\[ D' = \left( \frac{2}{3} \cdot 9, \frac{2}{3} \cdot 6 \right) \]
Calculating the x-coordinate:
\[ \frac{2}{3} \cdot 9 = \frac{18}{3} = 6 \]
Calculating the y-coordinate:
\[ \frac{2}{3} \cdot 6 = \frac{12}{3} = 4 \]
Thus, the coordinates of point \( D' \) are:
\[ D' = (6, 4) \]
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