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:
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Identify the coordinates of points \( D \) and \( F \). Let's assume:
- \( D = (x_D, y_D) \)
- \( F = (x_F, y_F) \)
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Calculate the length of segment \( DF \) using the distance formula: \[ DF = \sqrt{(x_F - x_D)^2 + (y_F - y_D)^2} \]
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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 \( Q \):
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The vector from \( D \) to \( F \) is given by: \[ \text{Vector } DF = (x_F - x_D, y_F - y_D) \]
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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: \[ Q = \left( 2x_F - x_D, 2y_F - y_D \right) \]
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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.