Question

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

Answer 1

To find point \( Q \) such that line \( DQ \) is an enlargement of line \( DF \) by a scale factor of 2 with point \( D \) as the center of dilation, we can use the formula for dilation.

The formula for a point \( F \) being dilated with respect to a center point \( D \) is given by:

\[
Q = D + k(F - D)
\]

where \( k \) is the scale factor of the dilation. Here, \( k = 2 \), \( D = (3,2) \), and \( F = (8,4) \).

First, we need to calculate the vector \( F - D \):

\[
F - D = (8 - 3, 4 - 2) = (5, 2)
\]

Next, we multiply this vector by the scale factor \( k \):

\[
k(F - D) = 2 \cdot (5, 2) = (10, 4)
\]

Now we add this result to point \( D \):

\[
Q = D + k(F - D) = (3, 2) + (10, 4) = (3 + 10, 2 + 4) = (13, 6)
\]

Therefore, the point \( Q \) is:

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