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To determine which point lies on a side of the original square RSTU, let's go through the steps needed to solve this problem:
### Step 1: Determine the translation vector
We are given the coordinates of [tex]\( S' \)[/tex] and [tex]\( S \)[/tex]:
- [tex]\( S' \)[/tex] has coordinates [tex]\( (-4, 1) \)[/tex]
- [tex]\( S \)[/tex] has coordinates [tex]\( (3, -5) \)[/tex]
We can find the translation vector by subtracting the coordinates of [tex]\( S' \)[/tex] from [tex]\( S \)[/tex]:
[tex]\[ \text{translation\_vector} = (S_x - S'_x, S_y - S'_y) = (3 - (-4), -5 - 1) = (7, -6) \][/tex]
This vector [tex]\((7, -6)\)[/tex] will translate any point on [tex]\( R'S'T'U' \)[/tex] to its corresponding point on [tex]\( RSTU \)[/tex].
### Step 2: Find the vertices of the pre-image square [tex]\( RSTU \)[/tex]
The vertices of [tex]\( R'S'T'U' \)[/tex] are:
- [tex]\( R'(-8, 1) \)[/tex]
- [tex]\( S'(-4, 1) \)[/tex]
- [tex]\( T'(-4, -3) \)[/tex]
- [tex]\( U'(-8, -3) \)[/tex]
Using the translation vector [tex]\((7, -6)\)[/tex], we translate each of these vertices to get the vertices of [tex]\( RSTU \)[/tex]:
- [tex]\( R = R' + \text{translation\_vector} = (-8 + 7, 1 - 6) = (-1, -5) \)[/tex]
- [tex]\( S = S' + \text{translation\_vector} = (-4 + 7, 1 - 6) = (3, -5) \)[/tex] (Confirming the given point [tex]\( S \)[/tex])
- [tex]\( T = T' + \text{translation\_vector} = (-4 + 7, -3 - 6) = (3, -9) \)[/tex]
- [tex]\( U = U' + \text{translation\_vector} = (-8 + 7, -3 - 6) = (-1, -9) \)[/tex]
### Step 3: Check which given points lie on a side of square [tex]\( RSTU \)[/tex]
We have the following points to check:
- [tex]\( (-5, -3) \)[/tex]
- [tex]\( (3, -3) \)[/tex]
- [tex]\( (-1, -6) \)[/tex]
- [tex]\( (4, -9) \)[/tex]
The vertices and sides of [tex]\( RSTU \)[/tex] are:
- [tex]\( R(-1, -5) \)[/tex]
- [tex]\( S(3, -5) \)[/tex]
- [tex]\( T(3, -9) \)[/tex]
- [tex]\( U(-1, -9) \)[/tex]
The sides of the square [tex]\( RSTU \)[/tex]:
- Side [tex]\( RS \)[/tex]: from [tex]\( (-1, -5) \)[/tex] to [tex]\( (3, -5) \)[/tex]
- Side [tex]\( ST \)[/tex]: from [tex]\( (3, -5) \)[/tex] to [tex]\( (3, -9) \)[/tex]
- Side [tex]\( TU \)[/tex]: from [tex]\( (3, -9) \)[/tex] to [tex]\( (-1, -9) \)[/tex]
- Side [tex]\( UR \)[/tex]: from [tex]\( (-1, -9) \)[/tex] to [tex]\( (-1, -5) \)[/tex]
Now, let's check each point to see if it lies on any sides of [tex]\( RSTU \)[/tex]:
- [tex]\( (-5, -3) \)[/tex] is not on any side of the square.
- [tex]\( (3, -3) \)[/tex] is not on any side of the square.
- [tex]\( (-1, -6) \)[/tex] is not on any side of the square.
- [tex]\( (4, -9) \)[/tex] is not on any side of the square.
None of these points lie on any side of the square [tex]\( RSTU \)[/tex].
### Conclusion
The point that lies on a side of the pre-image, square [tex]\( RSTU \)[/tex], cannot be any of the given choices. Thus, none of the given points lies on a side of the original square [tex]\( RSTU \)[/tex].
### Step 1: Determine the translation vector
We are given the coordinates of [tex]\( S' \)[/tex] and [tex]\( S \)[/tex]:
- [tex]\( S' \)[/tex] has coordinates [tex]\( (-4, 1) \)[/tex]
- [tex]\( S \)[/tex] has coordinates [tex]\( (3, -5) \)[/tex]
We can find the translation vector by subtracting the coordinates of [tex]\( S' \)[/tex] from [tex]\( S \)[/tex]:
[tex]\[ \text{translation\_vector} = (S_x - S'_x, S_y - S'_y) = (3 - (-4), -5 - 1) = (7, -6) \][/tex]
This vector [tex]\((7, -6)\)[/tex] will translate any point on [tex]\( R'S'T'U' \)[/tex] to its corresponding point on [tex]\( RSTU \)[/tex].
### Step 2: Find the vertices of the pre-image square [tex]\( RSTU \)[/tex]
The vertices of [tex]\( R'S'T'U' \)[/tex] are:
- [tex]\( R'(-8, 1) \)[/tex]
- [tex]\( S'(-4, 1) \)[/tex]
- [tex]\( T'(-4, -3) \)[/tex]
- [tex]\( U'(-8, -3) \)[/tex]
Using the translation vector [tex]\((7, -6)\)[/tex], we translate each of these vertices to get the vertices of [tex]\( RSTU \)[/tex]:
- [tex]\( R = R' + \text{translation\_vector} = (-8 + 7, 1 - 6) = (-1, -5) \)[/tex]
- [tex]\( S = S' + \text{translation\_vector} = (-4 + 7, 1 - 6) = (3, -5) \)[/tex] (Confirming the given point [tex]\( S \)[/tex])
- [tex]\( T = T' + \text{translation\_vector} = (-4 + 7, -3 - 6) = (3, -9) \)[/tex]
- [tex]\( U = U' + \text{translation\_vector} = (-8 + 7, -3 - 6) = (-1, -9) \)[/tex]
### Step 3: Check which given points lie on a side of square [tex]\( RSTU \)[/tex]
We have the following points to check:
- [tex]\( (-5, -3) \)[/tex]
- [tex]\( (3, -3) \)[/tex]
- [tex]\( (-1, -6) \)[/tex]
- [tex]\( (4, -9) \)[/tex]
The vertices and sides of [tex]\( RSTU \)[/tex] are:
- [tex]\( R(-1, -5) \)[/tex]
- [tex]\( S(3, -5) \)[/tex]
- [tex]\( T(3, -9) \)[/tex]
- [tex]\( U(-1, -9) \)[/tex]
The sides of the square [tex]\( RSTU \)[/tex]:
- Side [tex]\( RS \)[/tex]: from [tex]\( (-1, -5) \)[/tex] to [tex]\( (3, -5) \)[/tex]
- Side [tex]\( ST \)[/tex]: from [tex]\( (3, -5) \)[/tex] to [tex]\( (3, -9) \)[/tex]
- Side [tex]\( TU \)[/tex]: from [tex]\( (3, -9) \)[/tex] to [tex]\( (-1, -9) \)[/tex]
- Side [tex]\( UR \)[/tex]: from [tex]\( (-1, -9) \)[/tex] to [tex]\( (-1, -5) \)[/tex]
Now, let's check each point to see if it lies on any sides of [tex]\( RSTU \)[/tex]:
- [tex]\( (-5, -3) \)[/tex] is not on any side of the square.
- [tex]\( (3, -3) \)[/tex] is not on any side of the square.
- [tex]\( (-1, -6) \)[/tex] is not on any side of the square.
- [tex]\( (4, -9) \)[/tex] is not on any side of the square.
None of these points lie on any side of the square [tex]\( RSTU \)[/tex].
### Conclusion
The point that lies on a side of the pre-image, square [tex]\( RSTU \)[/tex], cannot be any of the given choices. Thus, none of the given points lies on a side of the original square [tex]\( RSTU \)[/tex].
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