Question

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

Answer 1

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

Given:
- Point \( D(3, 2) \)
- Point \( F(8, 4) \)

**Step 1: Find the vector from D to F.**

The vector \( \overrightarrow{DF} \) can be calculated as:
\[
\overrightarrow{DF} = F - D = (8 - 3, 4 - 2) = (5, 2)
\]

**Step 2: Apply the scale factor to the vector.**

The scale factor is 2, meaning that the vector \( \overrightarrow{DQ} \) will be twice the length of \( \overrightarrow{DF} \):
\[
\overrightarrow{DQ} = 2 \cdot \overrightarrow{DF} = 2 \cdot (5, 2) = (10, 4)
\]

**Step 3: Find the coordinates of point Q.**

To find the coordinates of \( Q \), we add the vector \( \overrightarrow{DQ} \) to point \( D \):
\[
Q = D + \overrightarrow{DQ} = (3, 2) + (10, 4) = (3 + 10, 2 + 4) = (13, 6)
\]

Therefore, the coordinates of point \( Q \) are:
\[
Q(13, 6)
\]

**Conclusion:**

The point \( Q(13, 6) \) is such that the segment \( DQ \) is an enlargement of the segment \( DF \) by a scale factor of 2, with point \( D(3, 2) \) as the center of dilation.