Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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].
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
