To find point H such that \( DH \) is an enlargement of \( DF \) by a scale factor of 2 with point D as the center of dilation, we can use the formula for dilation.
The formula for point \( P' \) after dilation from point \( P \) using a scale factor \( k \) and center of dilation \( D \) is given by:
\[ P' = D + k(P - D) \]
Here,
- \( D = (3, 2) \)
- \( F = (8, 4) \)
- \( k = 2 \)
First, we need to find the coordinates of point \( F \) relative to the center of dilation \( D \):
\[ F - D = (8 - 3, 4 - 2) = (5, 2) \]
Now we apply the scale factor \( k = 2 \):
\[ H - D = 2(F - D) = 2(5, 2) = (10, 4) \]
Now we add back the coordinates of point \( D \):
\[ H = D + (H - D) = (3, 2) + (10, 4) = (3 + 10, 2 + 4) = (13, 6) \]
Thus, the coordinates of point \( H \) are \( (13, 6) \).
To summarize:
The point \( H \) such that \( DH \) is an enlargement of \( DF \) by a scale factor of 2 with point D as the center of dilation is:
\[ \boxed{(13, 6)} \]