Sure, let's work through the problem step by step to find which point lies on a side of the pre-image, square RSTU.
1. Identify the Translation Vector:
The point [tex]\( S \)[/tex] corresponds to [tex]\( S' \)[/tex] after the square RSTU is translated. We are given the coordinates of [tex]\( S \)[/tex] and [tex]\( S' \)[/tex]:
[tex]\[
S = (3, -5)
\][/tex]
[tex]\[
S' = (-4, 1)
\][/tex]
The translation vector [tex]\( \vec{v} \)[/tex] is the difference between [tex]\( S' \)[/tex] and [tex]\( S \)[/tex]:
[tex]\[
\vec{v} = (S' - S) = (-4 - 3, 1 - (-5)) = (-7, 6)
\][/tex]
2. Find the Original Vertices R, T, U:
Using the translation vector [tex]\( \vec{v} = (-7, 6) \)[/tex], we can find the original coordinates of the vertices by reversing the translation on [tex]\( R' \)[/tex], [tex]\( T' \)[/tex], and [tex]\( U' \)[/tex].
For [tex]\( R \)[/tex]:
[tex]\[
R' = (-8, 1)
\][/tex]
[tex]\[
R = (R' - \vec{v}) = (-8 - (-7), 1 - 6) = (-8 + 7, 1 - 6) = (-1, -5)
\][/tex]
For [tex]\( S \)[/tex] (We already know it):
[tex]\[
S = (3, -5)
\][/tex]
For [tex]\( T \)[/tex]:
[tex]\[
T' = (-4, -3)
\][/tex]
[tex]\[
T = (T' - \vec{v}) = (-4 - (-7), -3 - 6) = (-4 + 7, -3 - 6) = (3, -9)
\][/tex]
For [tex]\( U \)[/tex]:
[tex]\[
U' = (-8, -3)
\][/tex]
[tex]\[
U = (U' - \vec{v}) = (-8 - (-7), -3 - 6) = (-8 + 7, -3 - 6) = (-1, -9)
\][/tex]
So, the pre-image vertices are:
[tex]\[
R(-1, -5), S(3, -5), T(3, -9), U(-1, -9)
\][/tex]
3. Examine Which Given Points Lie on Sides of Square RSTU:
We need to check the given points: [tex]\((-5, -3)\)[/tex], [tex]\((3, -3)\)[/tex], [tex]\((-1, -6)\)[/tex], and [tex]\((4, -9)\)[/tex].
Let's identify the sides of the square RSTU:
[tex]\[
\text{Side RS: } (R(-1,-5) \text{ to } S(3, -5))
\][/tex]
[tex]\[
\text{Side ST: } (S(3, -5) \text{ to } T(3, -9))
\][/tex]
[tex]\[
\text{Side TU: } (T(3, -9) \text{ to } U(-1, -9))
\][/tex]
[tex]\[
\text{Side UR: } (U(-1, -9) \text{ to } R(-1, -5))
\][/tex]
Checking each point:
- [tex]\((-5, -3)\)[/tex]: Does not lie on any of the sides RS, ST, TU, or UR.
- [tex]\((3, -3)\)[/tex]: Does not lie on any of the sides RS, ST, TU, or UR.
- [tex]\((-1, -6)\)[/tex]: Lies on side UR, as it aligns horizontally between [tex]\(U(-1, -9)\)[/tex] and [tex]\(R(-1, -5)\)[/tex]. Specifically:
[tex]\[
-9 \leq -6 \leq -5
\][/tex]
- [tex]\((4, -9)\)[/tex]: Does not lie on any of the sides RS, ST, TU, or UR.
4. Conclusion:
The point [tex]\((-1, -6)\)[/tex] lies on the side of the pre-image, square RSTU.
Final Answer:
[tex]\[
(-1, -6)
\][/tex]
