Answer :
To determine which point lies on a side of the pre-image square RSTU, we need to follow these steps:
1. Identify the Translation Vector:
We are given the coordinates of [tex]\(S\)[/tex] in the original square RSTU, which are [tex]\(S = (3, -5)\)[/tex]. The corresponding translated point is [tex]\(S' = (-4, 1)\)[/tex].
The translation vector can be calculated as the difference between the coordinates of [tex]\(S'\)[/tex] and [tex]\(S\)[/tex]:
[tex]\[ \text{Translation Vector} = (S'_{x} - S_{x}, S'_{y} - S_{y}) = (-4 - 3, 1 - (-5)) = (-7, 6) \][/tex]
2. Calculate the Coordinates of the Original Vertices:
By subtracting the translation vector from the vertices of the translated square [tex]\(R', S', T', U'\)[/tex], we can find the corresponding vertices in the original square RSTU.
- For [tex]\(R\)[/tex]:
[tex]\[ R = (R'_{x} - \text{Translation Vector}_{x}, R'_{y} - \text{Translation Vector}_{y}) = (-8 - (-7), 1 - 6) = (-1, -5) \][/tex]
- For [tex]\(T\)[/tex]:
[tex]\[ T = (T'_{x} - \text{Translation Vector}_{x}, T'_{y} - \text{Translation Vector}_{y}) = (-4 - (-7), -3 - 6) = (3, -9) \][/tex]
- For [tex]\(U\)[/tex]:
[tex]\[ U = (U'_{x} - \text{Translation Vector}_{x}, U'_{y} - \text{Translation Vector}_{y}) = (-8 - (-7), -3 - 6) = (-1, -9) \][/tex]
3. Check Each Point Against the Sides of the Original Square RSTU:
We need to check the given points [tex]\((-5, -3), (3, -3), (-1, -6), (4, -9)\)[/tex] to see if any of them lie on a side of the square RSTU.
- [tex]\((-5, -3)\)[/tex]:
- This point does not lie on any side of the square. It doesn't satisfy the equations of the sides formed by the vertices [tex]\((R, S), (S, T), (T, U), (U, R)\)[/tex].
- [tex]\((3, -3)\)[/tex]:
- This point does not lie on any side of the square. It doesn't satisfy the equations of the sides formed by the vertices [tex]\((R, S), (S, T), (T, U), (U, R)\)[/tex].
- [tex]\((-1, -6)\)[/tex]:
- This point does not lie on any side of the square. It doesn't satisfy the equations of the sides formed by the vertices [tex]\((R, S), (S, T), (T, U), (U, R)\)[/tex].
- [tex]\((4, -9)\)[/tex]:
- This point does not lie on any side of the square. It doesn't satisfy the equations of the sides formed by the vertices [tex]\((R, S), (S, T), (T, U), (U, R)\)[/tex].
Given the above analysis, none of the points [tex]\((-5, -3), (3, -3), (-1, -6), (4, -9)\)[/tex] lie on a side of the square RSTU.
Hence, the solution is:
[tex]\[ \boxed{\text{None of these points lie on a side of the pre-image, square RSTU.}} \][/tex]
1. Identify the Translation Vector:
We are given the coordinates of [tex]\(S\)[/tex] in the original square RSTU, which are [tex]\(S = (3, -5)\)[/tex]. The corresponding translated point is [tex]\(S' = (-4, 1)\)[/tex].
The translation vector can be calculated as the difference between the coordinates of [tex]\(S'\)[/tex] and [tex]\(S\)[/tex]:
[tex]\[ \text{Translation Vector} = (S'_{x} - S_{x}, S'_{y} - S_{y}) = (-4 - 3, 1 - (-5)) = (-7, 6) \][/tex]
2. Calculate the Coordinates of the Original Vertices:
By subtracting the translation vector from the vertices of the translated square [tex]\(R', S', T', U'\)[/tex], we can find the corresponding vertices in the original square RSTU.
- For [tex]\(R\)[/tex]:
[tex]\[ R = (R'_{x} - \text{Translation Vector}_{x}, R'_{y} - \text{Translation Vector}_{y}) = (-8 - (-7), 1 - 6) = (-1, -5) \][/tex]
- For [tex]\(T\)[/tex]:
[tex]\[ T = (T'_{x} - \text{Translation Vector}_{x}, T'_{y} - \text{Translation Vector}_{y}) = (-4 - (-7), -3 - 6) = (3, -9) \][/tex]
- For [tex]\(U\)[/tex]:
[tex]\[ U = (U'_{x} - \text{Translation Vector}_{x}, U'_{y} - \text{Translation Vector}_{y}) = (-8 - (-7), -3 - 6) = (-1, -9) \][/tex]
3. Check Each Point Against the Sides of the Original Square RSTU:
We need to check the given points [tex]\((-5, -3), (3, -3), (-1, -6), (4, -9)\)[/tex] to see if any of them lie on a side of the square RSTU.
- [tex]\((-5, -3)\)[/tex]:
- This point does not lie on any side of the square. It doesn't satisfy the equations of the sides formed by the vertices [tex]\((R, S), (S, T), (T, U), (U, R)\)[/tex].
- [tex]\((3, -3)\)[/tex]:
- This point does not lie on any side of the square. It doesn't satisfy the equations of the sides formed by the vertices [tex]\((R, S), (S, T), (T, U), (U, R)\)[/tex].
- [tex]\((-1, -6)\)[/tex]:
- This point does not lie on any side of the square. It doesn't satisfy the equations of the sides formed by the vertices [tex]\((R, S), (S, T), (T, U), (U, R)\)[/tex].
- [tex]\((4, -9)\)[/tex]:
- This point does not lie on any side of the square. It doesn't satisfy the equations of the sides formed by the vertices [tex]\((R, S), (S, T), (T, U), (U, R)\)[/tex].
Given the above analysis, none of the points [tex]\((-5, -3), (3, -3), (-1, -6), (4, -9)\)[/tex] lie on a side of the square RSTU.
Hence, the solution is:
[tex]\[ \boxed{\text{None of these points lie on a side of the pre-image, square RSTU.}} \][/tex]
