To solve the problem, we need to determine which of the given points lies on a side of the pre-image square RSTU.
Here are the steps:
1. Identify the Translation Vector:
Calculate the translation vector used to translate point [tex]\( S \)[/tex] with coordinates [tex]\( (3, -5) \)[/tex] to [tex]\( S' \)[/tex] with coordinates [tex]\( (-4, 1) \)[/tex].
Translation vector [tex]\( \vec{v} \)[/tex] can be found by subtracting the coordinates of [tex]\( S \)[/tex] from [tex]\( S' \)[/tex]:
[tex]\[
\vec{v} = (-4 - 3, 1 - (-5))
\][/tex]
[tex]\[
\vec{v} = (-4 - 3, 1 + 5)
\][/tex]
[tex]\[
\vec{v} = (-7, 6)
\][/tex]
2. Determine the Pre-Image Points:
Translate each of the given points of the square R'S'T'U' back using the inverse of the translation vector, [tex]\( -\vec{v} \)[/tex].
- For [tex]\( R'(-8, 1) \)[/tex]:
[tex]\[
R = (-8 + 7, 1 - 6) = (-1, -5)
\][/tex]
- For [tex]\( S'(-4, 1) \)[/tex]:
[tex]\[
S = (-4 + 7, 1 - 6) = (3, -5)
\][/tex]
- For [tex]\( T'(-4, -3) \)[/tex]:
[tex]\[
T = (-4 + 7, -3 - 6) = (3, -9)
\][/tex]
- For [tex]\( U'(-8, -3) \)[/tex]:
[tex]\[
U = (-8 + 7, -3 - 6) = (-1, -9)
\][/tex]
Therefore, the pre-image points (vertices of square RSTU) are:
[tex]\[
R = (-1, -5), S = (3, -5), T = (3, -9), U = (-1, -9)
\][/tex]
3. Identify the Correct Point:
Now, we need to check which of the given points lies on the side of the pre-image square RSTU. The given points are:
[tex]\((-5, -3)\)[/tex], [tex]\((3, -3)\)[/tex], [tex]\((-1, -6)\)[/tex], [tex]\((4, -9)\)[/tex].
Let's examine each point:
- [tex]\((-5, -3)\)[/tex]: This point does not match any side of the square RSTU.
- [tex]\((3, -3)\)[/tex]: This point does not match any side of the square RSTU.
- [tex]\((-1, -6)\)[/tex]: This point does not match any side of the square RSTU.
- [tex]\((4, -9)\)[/tex]: This point does not match any side of the square RSTU.
None of these points correspond to any side of the pre-image square RSTU, and therefore, none of the provided points lie on the side of the pre-image square.
So, the final answer is:
[tex]\[
\boxed{\text{None}}
\][/tex]
