Bot GPT 4-o mini To find the point Q such that the line segment \( DQ \) is an enlargement of the line segment \( DF \) by a scale factor of 2, with D as the center of dilation, you need to follow these steps:
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Identify the Coordinates of D and F: You will need 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 Vector DF: This vector can be found using: \[ DF = (x_F - x_D, y_F - y_D) \]
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Enlargement by Scale Factor: To enlarge the vector \( DF \) by a scale factor of 2, you multiply the vector by 2: \[ DQ = 2 \cdot DF = 2 \cdot (x_F - x_D, y_F - y_D) = (2(x_F - x_D), 2(y_F - y_D)) \]
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Find the Coordinates of Q: You can find the coordinates of Q by adding the vector \( DQ \) to the coordinates of D: \[ Q = D + DQ = (x_D + 2(x_F - x_D), y_D + 2(y_F - y_D)) \]
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Final Result: Substitute \( x_D, y_D, x_F, y_F \) into the formula to get the coordinates of Q.
For instance, if you assume \( D(1, 2) \) and \( F(3, 4) \):
- DF would be \( (3 - 1, 4 - 2) = (2, 2) \)
- Enlarged \( DQ \) would be \( (2 \cdot 2, 2 \cdot 2) = (4, 4) \)
- So, \( Q \) would be \( (1 + 4, 2 + 4) = (5, 6) \)
Thus, in this example, Q is \( (5, 6) \).
You need to replace the actual values of D and F to find the specific coordinates of Q.
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