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. (1 point) Q ( ,)

D=(3,2) F=(8,4)

Answer 1

To find point Q such that line segment \( \overline{DQ} \) is an enlargement of line segment \( \overline{DF} \) by a scale factor of 2 with point D as the center of dilation, we can use the following steps:

1. **Determine the vector from D to F**:
\( D = (3, 2) \) and \( F = (8, 4) \).
The vector \( \overrightarrow{DF} \) can be calculated as:
\[
\overrightarrow{DF} = F - D = (8 - 3, 4 - 2) = (5, 2).
\]

2. **Scale the vector**:
Since we need an enlargement by a scale factor of 2, we multiply the vector \( \overrightarrow{DF} \) by 2:
\[
2 \cdot \overrightarrow{DF} = 2 \cdot (5, 2) = (10, 4).
\]

3. **Find point Q**:
To find point Q, we add this scaled vector to point D:
\[
Q = D + 2 \cdot \overrightarrow{DF} = (3, 2) + (10, 4) = (3 + 10, 2 + 4) = (13, 6).
\]

Therefore, point Q is \( \boxed{(13, 6)} \).