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Sagot :
To determine the new [tex]\(y\)[/tex]-coordinate when a translation of [tex]\(T_{-3,-8}(x, y)\)[/tex] is applied to point [tex]\(B\)[/tex] of square [tex]\(ABCD\)[/tex], we need to understand how the translation affects the [tex]\(y\)[/tex]-coordinate.
Given the translation [tex]\(T_{-3, -8}(x, y)\)[/tex], this means every point [tex]\((x, y)\)[/tex] on the square is moved 3 units to the left (affecting [tex]\(x\)[/tex]) and 8 units down (affecting [tex]\(y\)[/tex]). The translation of [tex]\((x, y)\)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\((x - 3, y - 8)\)[/tex].
Our specific task is to find out which of the given initial [tex]\(y\)[/tex]-coordinates of point [tex]\(B\)[/tex] correctly results in a certain translated [tex]\(y\)[/tex]-coordinate after applying the translation.
Let's look at the given options for the initial [tex]\(y\)[/tex]-coordinate of [tex]\(B\)[/tex]:
1. [tex]\(-12\)[/tex]
2. [tex]\(10\)[/tex]
3. [tex]\(-6\)[/tex]
4. [tex]\(-2\)[/tex]
We apply the translation step-by-step calculation for each potential initial [tex]\(y\)[/tex]-coordinate to see what the new [tex]\(y\)[/tex]-coordinate of [tex]\(B\)[/tex] would be after translation:
1. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(-12\)[/tex]
[tex]\[ y_B = -12 - 8 = -20 \][/tex]
2. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(10\)[/tex]
[tex]\[ y_B = 10 - 8 = 2 \][/tex]
3. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(-6\)[/tex]
[tex]\[ y_B = -6 - 8 = -14 \][/tex]
4. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(-2\)[/tex]
[tex]\[ y_B = -2 - 8 = -10 \][/tex]
After applying the translation to each of these given initial [tex]\(y\)[/tex]-coordinates, we get the results:
- [tex]\(-12\)[/tex] translates to [tex]\(-20\)[/tex]
- [tex]\(10\)[/tex] translates to [tex]\(2\)[/tex]
- [tex]\(-6\)[/tex] translates to [tex]\(-14\)[/tex]
- [tex]\(-2\)[/tex] translates to [tex]\(-10\)[/tex]
With this detailed step-by-step solution, we see that the new [tex]\(y\)[/tex]-coordinate would clearly depend on which initial value is being translated. This confirms that we have correctly applied the translation [tex]\(T_{-3,-8}(x, y)\)[/tex]. Given the options, we can analyze if any specific condition needs to be met for a typical [tex]\(y\)[/tex]-coordinate in the problem context. However, if considering the output for just the resultant position due to translation, these translations illustrate the mathematical application of translations to each option.
Given the translation [tex]\(T_{-3, -8}(x, y)\)[/tex], this means every point [tex]\((x, y)\)[/tex] on the square is moved 3 units to the left (affecting [tex]\(x\)[/tex]) and 8 units down (affecting [tex]\(y\)[/tex]). The translation of [tex]\((x, y)\)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\((x - 3, y - 8)\)[/tex].
Our specific task is to find out which of the given initial [tex]\(y\)[/tex]-coordinates of point [tex]\(B\)[/tex] correctly results in a certain translated [tex]\(y\)[/tex]-coordinate after applying the translation.
Let's look at the given options for the initial [tex]\(y\)[/tex]-coordinate of [tex]\(B\)[/tex]:
1. [tex]\(-12\)[/tex]
2. [tex]\(10\)[/tex]
3. [tex]\(-6\)[/tex]
4. [tex]\(-2\)[/tex]
We apply the translation step-by-step calculation for each potential initial [tex]\(y\)[/tex]-coordinate to see what the new [tex]\(y\)[/tex]-coordinate of [tex]\(B\)[/tex] would be after translation:
1. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(-12\)[/tex]
[tex]\[ y_B = -12 - 8 = -20 \][/tex]
2. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(10\)[/tex]
[tex]\[ y_B = 10 - 8 = 2 \][/tex]
3. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(-6\)[/tex]
[tex]\[ y_B = -6 - 8 = -14 \][/tex]
4. Initial [tex]\(y\)[/tex]-coordinate: [tex]\(-2\)[/tex]
[tex]\[ y_B = -2 - 8 = -10 \][/tex]
After applying the translation to each of these given initial [tex]\(y\)[/tex]-coordinates, we get the results:
- [tex]\(-12\)[/tex] translates to [tex]\(-20\)[/tex]
- [tex]\(10\)[/tex] translates to [tex]\(2\)[/tex]
- [tex]\(-6\)[/tex] translates to [tex]\(-14\)[/tex]
- [tex]\(-2\)[/tex] translates to [tex]\(-10\)[/tex]
With this detailed step-by-step solution, we see that the new [tex]\(y\)[/tex]-coordinate would clearly depend on which initial value is being translated. This confirms that we have correctly applied the translation [tex]\(T_{-3,-8}(x, y)\)[/tex]. Given the options, we can analyze if any specific condition needs to be met for a typical [tex]\(y\)[/tex]-coordinate in the problem context. However, if considering the output for just the resultant position due to translation, these translations illustrate the mathematical application of translations to each option.
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