Question

Find point Q to prove that DQ¯¯¯¯¯¯¯¯
is an enlargement of DF¯¯¯¯¯¯¯¯
by a scale factor of 2 with a point D as the center of dilation.

Answer 1

To prove that line segment \( DQ \) is an enlargement of line segment \( DF \) by a scale factor of 2 with point \( D \) as the center of dilation, we need to find point \( Q \) such that the length of segment \( DQ \) is twice the length of segment \( DF \).

### Given:
- \( D \) is the center of dilation.
- \( DF \) is the original segment.
- The scale factor \( k \) is \( 2 \).

### Steps:

1. **Identify the coordinates** of points \( D \) and \( F \). Let's assume:
- \( D = (x_D, y_D) \)
- \( F = (x_F, y_F) \)

2. **Calculate the length of segment \( DF \)** using the distance formula:
\[
DF = \sqrt{(x_F - x_D)^2 + (y_F - y_D)^2}
\]

3. **Determine the coordinates of point \( Q \)**: Since \( Q \) must be in the same direction from \( D \) as \( F \) and be twice as far, we can calculate the coordinates of \( Q \):
- The vector from \( D \) to \( F \) is given by:
\[
\text{Vector } DF = (x_F - x_D, y_F - y_D)
\]

- To find \( Q \), extend this vector by scale factor \( 2 \):
\[
Q = D + 2 \cdot \text{Vector } DF
\]
Thus,
\[
Q = \left( x_D + 2 \cdot (x_F - x_D), y_D + 2 \cdot (y_F - y_D) \right)
\]
Simplifying this gives:
\[
Q = \left( 2x_F - x_D, 2y_F - y_D \right)
\]

### Conclusion:
Point \( Q \) is given by the coordinates \( Q = (2x_F - x_D, 2y_F - y_D) \).

To confirm:
- The length \( DQ \) should be:
\[
DQ = \sqrt{(2x_F - x_D - x_D)^2 + (2y_F - y_D - y_D)^2}
\]
Simplifying, we will find that:
\[
DQ = 2DF
\]

Thus, \( DQ \) is indeed an enlargement of \( DF \) by a scale factor of 2, confirming that \( DQ \) is proportional to \( DF \) with \( D \) as the center of dilation.