To prove that the line segment \( \overline{DQ} \) is an enlargement of \( \overline{DF} \) by a scale factor of 2 with point \( D \) as the center of dilation, you need to first clarify the coordinates of points \( D \), \( F \), and \( Q \).
Let's assume the coordinates of points \( D \) and \( F \) are as follows:
- \( D(x_D, y_D) \)
- \( F(x_F, y_F) \)
To find point \( Q \) such that \( \overline{DQ} \) is an enlargement of \( \overline{DF} \) by a scale factor of 2, we can use the formula for dilation.
Steps to find point \( Q \):
-
Coordinates of Dilation: The formula for dilation of a point \( (x, y) \) with respect to a center point \( (x_D, y_D) \) and a scale factor \( k \) is given by: \[ (x', y') = (x_D + k \cdot (x - x_D), y_D + k \cdot (y - y_D)) \] where \( (x', y') \) are the coordinates of the dilated point.
-
Apply the Dilation Formula: In our case, since \( k = 2 \) for the enlargement, we can compute the coordinates of point \( Q \) as follows: \[ (x_Q, y_Q) = (x_D + 2 \cdot (x_F - x_D), y_D + 2 \cdot (y_F - y_D)) \]
-
Calculate Specifically (if necessary): If you have specific coordinates for \( D \) and \( F \), substitute those into the formulas above to find \( Q \).
-
Verification: After calculating \( Q \), you can verify that \( \overline{DQ} \) is indeed twice the length of \( \overline{DF} \) by measuring the distance between points, ensuring that: \[ \text{Distance } DF = \sqrt{(x_F - x_D)^2 + (y_F - y_D)^2} \] \[ \text{Distance } DQ = \sqrt{(x_Q - x_D)^2 + (y_Q - y_D)^2} \] Then check if \( DQ = 2 \times DF \).
Example:
Let’s assume:
- \( D(2, 3) \)
- \( F(4, 5) \)
Using the dilation formula to find \( Q \):
-
Calculate \( Q \): \[ x_Q = 2 + 2 \cdot (4 - 2) = 2 + 2 \cdot 2 = 6 \] \[ y_Q = 3 + 2 \cdot (5 - 3) = 3 + 2 \cdot 2 = 7 \] So \( Q(6, 7) \).
-
Check the distances:
- Distance \( DF \): \[ DF = \sqrt{(4 - 2)^2 + (5 - 3)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
- Distance \( DQ \): \[ DQ = \sqrt{(6 - 2)^2 + (7 - 3)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
- Verify: \[ DQ = 2 \times DF \rightarrow 4\sqrt{2} = 2 \times 2\sqrt{2} \text{ (True)} \]
Therefore, point \( Q \) is correctly calculated. Please provide coordinates for \( D \) and \( F \) if you need specific calculations!