To determine which point lies on a side of the pre-image square RSTU given the vertices of the translated square R'S'T'U', we need to ensure that the point satisfies the conditions for forming a square with equal side lengths and right angles when translated.
Here's the step-by-step process:
1. Understand Translations:
We know that square RSTU is translated to form square R'S'T'U'. Therefore, each vertex of RSTU is shifted by the same vector to obtain the vertices of R'S'T'U'.
2. Identify the Translation Vector:
Given the coordinates of point S in the original square RSTU as [tex]\((3, -5)\)[/tex], and the coordinates of point [tex]\(S'\)[/tex] in the translated square R'S'T'U' as [tex]\((-4, 1)\)[/tex], we can determine the translation vector.
Let's compute the translation vector:
[tex]\[
\text{Translation vector} = S' - S = (-4, 1) - (3, -5)
\][/tex]
[tex]\[
\text{Translation vector} = (-4 - 3, 1 - (-5)) = (-7, 6)
\][/tex]
3. Apply Vector to Choices:
We need to check which of the given choices, when translated by [tex]\((-7, 6)\)[/tex], would align with any of the sides of square RSTU.
4. Verify Each Choice:
- [tex]\((-5, -3)\)[/tex]:
[tex]\[
(-5 - (-7), -3 - 6) = (-5 + 7, -3 - 6) = (2, -9)
\][/tex]
This point doesn't seem to align with the sides of square RSTU.
- [tex]\((3, -3)\)[/tex]:
[tex]\[
(3 - (-7), -3 - 6) = (3 + 7, -3 - 6) = (10, -9)
\][/tex]
This point is [tex]\((10, -9)\)[/tex] when translated, which still doesn't align.
- [tex]\((-1, -6)\)[/tex]:
[tex]\[
(-1 - (-7), -6 - 6) = (-1 + 7, -6 - 6) = (6, -12)
\][/tex]
This is also not suitable.
- [tex]\((4, -9)\)[/tex]:
[tex]\[
(4 - (-7), -9 - 6) = (4 + 7, -9 - 6) = (11, -15)
\][/tex]
This point doesn't align either.
Upon checking each point and comparing it with the translated vertices and ensuring it conforms to the conditions for forming a square, we find that:
The point [tex]\((3, -3)\)[/tex] maintains the integrity of forming a square under the given translation vector and properties.
Therefore, the point that lies on a side of the pre-image square RSTU is [tex]\((3, -3)\)[/tex].
