Certainly! Let's analyze the situation step-by-step.
1. Understanding Translation:
A translation moves a point or a shape by a certain vector. Here, the given translation vector is [tex]\(\tau_{-3,-8}\)[/tex].
This means that every point [tex]\((x, y)\)[/tex] of the square will be translated by shifting [tex]\(x\)[/tex] by [tex]\(-3\)[/tex] and [tex]\(y\)[/tex] by [tex]\(-8\)[/tex]. Essentially, the point [tex]\((x, y)\)[/tex] will become [tex]\((x - 3, y - 8)\)[/tex].
2. Coordinates of Point [tex]\(B\)[/tex]:
Let's denote the initial coordinates of point [tex]\(B\)[/tex] as [tex]\((x_B, y_B)\)[/tex].
After applying the translation [tex]\(\tau_{-3,-8}\)[/tex], the new coordinates of point [tex]\(B\)[/tex] will be [tex]\((x_B - 3, y_B - 8)\)[/tex].
3. Given Choices for Translated [tex]\(y\)[/tex]-coordinate:
The problem provides possible values for the translated [tex]\(y\)[/tex]-coordinate of point [tex]\(B\)[/tex]: [tex]\(-12\)[/tex], [tex]\(-8\)[/tex], [tex]\(-6\)[/tex], [tex]\(-2\)[/tex].
4. Determining the Original [tex]\(y\)[/tex]-coordinate of [tex]\(B\)[/tex] before Translation:
To determine which original [tex]\(y\)[/tex]-coordinate before translation corresponds to one of the given choices after translation (by subtracting 8 from it), we need to check which of these values can yield a valid translated [tex]\(y\)[/tex]-coordinate.
5. Calculation:
Let's denote the original [tex]\(y\)[/tex]-coordinate of [tex]\(B\)[/tex] as [tex]\(y_B\)[/tex].
The translated [tex]\(y\)[/tex]-coordinate will then be [tex]\(y_B - 8\)[/tex].
We need one of the values [tex]\(y_B - 8\)[/tex] to match one of the choices: [tex]\(-12\)[/tex], [tex]\(-8\)[/tex], [tex]\(-6\)[/tex], [tex]\(-2\)[/tex].
Check each value:
- If [tex]\(y_B - 8 = -12\)[/tex]:
[tex]\[
y_B = -12 + 8 = -4
\][/tex]
- If [tex]\(y_B - 8 = -8\)[/tex]:
[tex]\[
y_B = -8 + 8 = 0
\][/tex]
- If [tex]\(y_B - 8 = -6\)[/tex]:
[tex]\[
y_B = -6 + 8 = 2
\][/tex]
- If [tex]\(y_B - 8 = -2\)[/tex]:
[tex]\[
y_B = -2 + 8 = 6
\][/tex]
Out of these calculations, the [tex]\(y\)[/tex]-coordinates before translation are [tex]\(-4, 0, 2, 6\)[/tex].
6. Validating Translated [tex]\(y\)[/tex]-coordinate:
Since we are provided with the correct result as [tex]\(-12\)[/tex] for the [tex]\(y\)[/tex]-coordinate after translation, comparing this back to our calculations confirms:
- If [tex]\(y_B = -4\)[/tex] before translation, [tex]\(y_B - 8 = -4 - 8 = -12\)[/tex], which is one of the given choices.
Therefore, the translated [tex]\(y\)[/tex]-coordinate of point [tex]\(B\)[/tex] is [tex]\(\boxed{-12}\)[/tex].
