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 (3,2) as the center of dilation.

(1 point)

Answer 1

To find the point \( Q \) such that the segment \( DQ \) is an enlargement of \( DF \) with a scale factor of 2 and \( D \) as the center of dilation, we can follow these steps:

1. Identify the coordinates of points \( D \) and \( F \):
- \( D(3, 2) \)
- \( F(8, 4) \)

2. Calculate the vector \( DF \):
- \( DF = F - D = (8 - 3, 4 - 2) = (5, 2) \)

3. Compute the coordinates of the point \( Q \) by applying the scale factor of 2 to the vector \( DF \):
- The scaled vector is \( 2 \cdot DF = 2 \cdot (5, 2) = (10, 4) \)

4. Add this scaled vector to the coordinates of point \( D \):
- \( Q = D + 2 \cdot DF = (3, 2) + (10, 4) = (3 + 10, 2 + 4) = (13, 6) \)

Thus, the coordinates of point \( Q \) are \( (13, 6) \).

The dilation \( DQ \) will be an enlargement of \( DF \) by a scale factor of 2, confirming that \( Q \) is indeed the point we were looking for.