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Sagot :
To determine the coordinates of the endpoints of the dilated segment [tex]\(\overline{A^{\prime} B^{\prime}}\)[/tex] when [tex]\(\overline{A B}\)[/tex] is dilated by a scale factor of 2 centered at (3,5), we need to follow these steps:
1. Identify the initial coordinates of the points and the center of dilation:
- [tex]\(A(-1, 6)\)[/tex]
- [tex]\(B(7, 4)\)[/tex]
- Center [tex]\((3, 5)\)[/tex]
2. Calculate the vectors from the center of dilation to each point:
- Vector from the center to point [tex]\(A\)[/tex]: [tex]\(A_{vec} = (-1, 6) - (3, 5) = (-1 - 3, 6 - 5) = (-4, 1)\)[/tex]
- Vector from the center to point [tex]\(B\)[/tex]: [tex]\(B_{vec} = (7, 4) - (3, 5) = (7 - 3, 4 - 5) = (4, -1)\)[/tex]
3. Scale these vectors by the given scale factor of 2:
- Scaled vector for [tex]\(A\)[/tex]: [tex]\(scaled\_A_{vec} = 2 \times (-4, 1) = (-8, 2)\)[/tex]
- Scaled vector for [tex]\(B\)[/tex]: [tex]\(scaled\_B_{vec} = 2 \times (4, -1) = (8, -2)\)[/tex]
4. Find the new coordinates by adding the scaled vectors back to the center of dilation:
- New coordinates for [tex]\(A^{\prime}\)[/tex]: [tex]\(A^{\prime} = (3, 5) + (-8, 2) = (3 - 8, 5 + 2) = (-5, 7)\)[/tex]
- New coordinates for [tex]\(B^{\prime}\)[/tex]: [tex]\(B^{\prime} = (3, 5) + (8, -2) = (3 + 8, 5 - 2) = (11, 3)\)[/tex]
5. Compare these new coordinates with the given options:
Let's check each option:
- Option (1) [tex]\(A^{\prime}=(-7,5)\)[/tex] and [tex]\(B^{\prime}=9,1)\)[/tex]
- Option (2) [tex]\(A^{\prime}=(-1,6)\)[/tex] and [tex]\(B^{\prime}=7,4)\)[/tex]
- Option (3) [tex]\(A^{\prime}=(-6,8)\)[/tex] and [tex]\(B^{\prime}=10,4)\)[/tex]
- Option (4) [tex]\(A^{\prime}=(-9,3)\)[/tex] and [tex]\(B^{\prime}=7,-1)\)[/tex]
From our calculations, we obtain [tex]\((-5, 7)\)[/tex] for [tex]\(A^{\prime}\)[/tex] and [tex]\((11, 3)\)[/tex] for [tex]\(B^{\prime}\)[/tex]. It is clear that these coordinates do not match any of the given options.
Therefore, the correct option is:
[tex]\[ \boxed{\text{None of the above}} \][/tex]
1. Identify the initial coordinates of the points and the center of dilation:
- [tex]\(A(-1, 6)\)[/tex]
- [tex]\(B(7, 4)\)[/tex]
- Center [tex]\((3, 5)\)[/tex]
2. Calculate the vectors from the center of dilation to each point:
- Vector from the center to point [tex]\(A\)[/tex]: [tex]\(A_{vec} = (-1, 6) - (3, 5) = (-1 - 3, 6 - 5) = (-4, 1)\)[/tex]
- Vector from the center to point [tex]\(B\)[/tex]: [tex]\(B_{vec} = (7, 4) - (3, 5) = (7 - 3, 4 - 5) = (4, -1)\)[/tex]
3. Scale these vectors by the given scale factor of 2:
- Scaled vector for [tex]\(A\)[/tex]: [tex]\(scaled\_A_{vec} = 2 \times (-4, 1) = (-8, 2)\)[/tex]
- Scaled vector for [tex]\(B\)[/tex]: [tex]\(scaled\_B_{vec} = 2 \times (4, -1) = (8, -2)\)[/tex]
4. Find the new coordinates by adding the scaled vectors back to the center of dilation:
- New coordinates for [tex]\(A^{\prime}\)[/tex]: [tex]\(A^{\prime} = (3, 5) + (-8, 2) = (3 - 8, 5 + 2) = (-5, 7)\)[/tex]
- New coordinates for [tex]\(B^{\prime}\)[/tex]: [tex]\(B^{\prime} = (3, 5) + (8, -2) = (3 + 8, 5 - 2) = (11, 3)\)[/tex]
5. Compare these new coordinates with the given options:
Let's check each option:
- Option (1) [tex]\(A^{\prime}=(-7,5)\)[/tex] and [tex]\(B^{\prime}=9,1)\)[/tex]
- Option (2) [tex]\(A^{\prime}=(-1,6)\)[/tex] and [tex]\(B^{\prime}=7,4)\)[/tex]
- Option (3) [tex]\(A^{\prime}=(-6,8)\)[/tex] and [tex]\(B^{\prime}=10,4)\)[/tex]
- Option (4) [tex]\(A^{\prime}=(-9,3)\)[/tex] and [tex]\(B^{\prime}=7,-1)\)[/tex]
From our calculations, we obtain [tex]\((-5, 7)\)[/tex] for [tex]\(A^{\prime}\)[/tex] and [tex]\((11, 3)\)[/tex] for [tex]\(B^{\prime}\)[/tex]. It is clear that these coordinates do not match any of the given options.
Therefore, the correct option is:
[tex]\[ \boxed{\text{None of the above}} \][/tex]
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