To determine point \( H \) such that line segment \( DH \) is an enlargement of line segment \( DF \) by a scale factor of 2 with center of dilation at point \( D(3,2) \), we start by finding the coordinates of point \( F(8,4) \) after applying the dilation.
The formula for finding the coordinates of a point after dilation about a center point is:
\[ H = D + k \cdot (F - D) \]
Here, \( D \) is the center of dilation, \( F \) is the original point, and \( k \) is the scale factor. Set \( k = 2 \) since we want to dilate by a scale factor of 2.
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First, find \( F - D \): \[ F - D = (8 - 3, 4 - 2) = (5, 2) \]
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Now, multiply by the scale factor \( k = 2 \): \[ k \cdot (F - D) = 2 \cdot (5, 2) = (10, 4) \]
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Then, add this vector to point \( D \): \[ H = D + k \cdot (F - D) = (3,2) + (10,4) = (3 + 10, 2 + 4) = (13, 6) \]
With this computation showing that \( H \) should be \( (13, 6) \), we can compare this with the given options:
- \( H \) is at \( (5.5, 3) \)
- \( H \) is at \( (8, 18) \)
- \( H \) is at \( (12, 6) \)
- \( H \) is at \( (-2, 0) \)
None of the options match \( (13, 6) \). However, we can still ascertain if any of them could be a candidate for some other form of transformation or if we miscalculate; we see that the closest point is \( (12, 6) \).
Let’s double-check the dilation calculation:
Now check if \( (12, 6) \) retains the property of being an enlargement from \( D \):
- Calculate \( (12, 6) - (3, 2) = (9, 4) \)
To verify the scale: \[ \text{Distance of } F \text{ to } D = \sqrt{(8 - 3)^2 + (4 - 2)^2} = \sqrt{25 + 4} = \sqrt{29} \] Distance of \( H (12, 6) \) to \( D (3, 2) \): \[ \text{Distance of } H \text{ to } D = \sqrt{(12 - 3)^2 + (6 - 2)^2} = \sqrt{81 + 16} = \sqrt{97} \]
The scale factor: \[ \frac{DH}{DF} = \frac{\sqrt{97}}{\sqrt{29}} \approx 1.9 \]
After my calculations, none match the actual \( DH \) being only twice \( DF \). Thus, it seems the task fundamentally has an error because none provided \( H \) can complete an enlargement by a precise scale of 2 from the derived properties.
Conclusion:
Given your choices, none were correct based on computations for dilation to meet dimensional criteria as prescribed. Adjust or re-evaluate points is advisable if allowed.
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