Question

Find point H to prove that DH is an enlargement of DF by a scale factor of 2 with point D as the center of dilation.
D (3,2)
F(8,4)

Answer 1

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:

\[
\boxed{(13, 6)}
\]