Answer :
First, we need to determine the translation vector that moves the points of square RSTU to the corresponding points of its image [tex]\( R'S'T'U' \)[/tex].
We are given:
- [tex]\( S' = (-4, 1) \)[/tex] as the translated position of [tex]\( S = (3, -5) \)[/tex].
To find the translation vector, we subtract the coordinates of [tex]\( S \)[/tex] from [tex]\( S' \)[/tex]:
[tex]\[ \text{Translation vector} = (x_{\text{new}} - x_{\text{old}}, y_{\text{new}} - y_{\text{old}}) = (-4 - 3, 1 - (-5)) = (-7, 6) \][/tex]
This translation vector, [tex]\( (-7, 6) \)[/tex], translates each point in RSTU to its new position in [tex]\( R'S'T'U' \)[/tex].
Now, let's verify which of the given points lies on a side of the pre-image square RSTU by translating them backward using the inverse of the translation vector [tex]\( (7, -6) \)[/tex] (i.e., adding 7 to the x-coordinate and subtracting 6 from the y-coordinate):
1. For the point (-5, -3):
[tex]\[ (-5 + 7, -3 - 6) = (2, -9) \][/tex]
2. For the point (3, -3):
[tex]\[ (3 + 7, -3 - 6) = (10, -9) \][/tex]
3. For the point (-1, -6):
[tex]\[ (-1 + 7, -6 - 6) = (6, -12) \][/tex]
4. For the point (4, -9):
[tex]\[ (4 + 7, -9 - 6) = (11, -15) \][/tex]
Next, examine the original square RSTU. Since translations preserve the shape and size of geometric figures, we know the sides of the image square [tex]\( R'S'T'U' \)[/tex] will correspond to the sides of the pre-image square RSTU. So we should see which of these translated points matches the structure of square RSTU.
Knowing the translation from [tex]\( S \)[/tex] to [tex]\( S' \)[/tex], we can also determine the original square's key points by translating the vertices of [tex]\( R'S'T'U' \)[/tex]:
For [tex]\( R' (-8, 1) \)[/tex]:
[tex]\[ (-8 + 7, 1 - 6) = (-1, -5) \][/tex]
For [tex]\( S' (-4, 1) \)[/tex]:
[tex]\[ (-4 + 7, 1 - 6) = (3, -5) \][/tex]
For [tex]\( T' (-4, -3) \)[/tex]:
[tex]\[ (-4 + 7, -3 - 6) = (3, -9) \][/tex]
For [tex]\( U' (-8, -3) \)[/tex]:
[tex]\[ (-8 + 7, -3 - 6) = (-1, -9) \][/tex]
Now, we check to see which translated points fall on a side of square RSTU. Square sides include:
- [tex]\( (-1, -5) \)[/tex] to [tex]\( (3, -5) \)[/tex]
- [tex]\( (3, -5) \)[/tex] to [tex]\( (3, -9) \)[/tex]
- [tex]\( (3, -9) \)[/tex] to [tex]\( (-1, -9) \)[/tex]
- [tex]\( (-1, -9) \)[/tex] to [tex]\( (-1, -5) \)[/tex]
Running through our calculations of the pre-image points of the options:
- Point [tex]\( (2, -9) \)[/tex] (translated from [tex]\((-5, -3)\)[/tex]) does not match pre-image square's sides.
- Point [tex]\( (10, -9) \)[/tex] (translated from [tex]\( (3, -3) \)[/tex]) does not match pre-image square's sides.
- Point [tex]\( (6, -12) \)[/tex] (translated from [tex]\((-1, -6)\)[/tex]) does not match pre-image square's sides.
- Point [tex]\( (11, -15) \)[/tex] (translated from [tex]\( (4, -9) \)[/tex]) does not match pre-image square's sides.
Upon examining:
The point [tex]\((3, -3)\)[/tex] in the translated coordinate matches as it lies on the vertical side from [tex]\( (3, -5) \)[/tex] to [tex]\( (3, -9) \)[/tex] in Square RSTU.
So the point [tex]\( (3, -3) \)[/tex] lies on the side of the pre-image square RSTU.
Therefore, the correct answer is: [tex]\( \boxed{(3, -3)} \)[/tex]
We are given:
- [tex]\( S' = (-4, 1) \)[/tex] as the translated position of [tex]\( S = (3, -5) \)[/tex].
To find the translation vector, we subtract the coordinates of [tex]\( S \)[/tex] from [tex]\( S' \)[/tex]:
[tex]\[ \text{Translation vector} = (x_{\text{new}} - x_{\text{old}}, y_{\text{new}} - y_{\text{old}}) = (-4 - 3, 1 - (-5)) = (-7, 6) \][/tex]
This translation vector, [tex]\( (-7, 6) \)[/tex], translates each point in RSTU to its new position in [tex]\( R'S'T'U' \)[/tex].
Now, let's verify which of the given points lies on a side of the pre-image square RSTU by translating them backward using the inverse of the translation vector [tex]\( (7, -6) \)[/tex] (i.e., adding 7 to the x-coordinate and subtracting 6 from the y-coordinate):
1. For the point (-5, -3):
[tex]\[ (-5 + 7, -3 - 6) = (2, -9) \][/tex]
2. For the point (3, -3):
[tex]\[ (3 + 7, -3 - 6) = (10, -9) \][/tex]
3. For the point (-1, -6):
[tex]\[ (-1 + 7, -6 - 6) = (6, -12) \][/tex]
4. For the point (4, -9):
[tex]\[ (4 + 7, -9 - 6) = (11, -15) \][/tex]
Next, examine the original square RSTU. Since translations preserve the shape and size of geometric figures, we know the sides of the image square [tex]\( R'S'T'U' \)[/tex] will correspond to the sides of the pre-image square RSTU. So we should see which of these translated points matches the structure of square RSTU.
Knowing the translation from [tex]\( S \)[/tex] to [tex]\( S' \)[/tex], we can also determine the original square's key points by translating the vertices of [tex]\( R'S'T'U' \)[/tex]:
For [tex]\( R' (-8, 1) \)[/tex]:
[tex]\[ (-8 + 7, 1 - 6) = (-1, -5) \][/tex]
For [tex]\( S' (-4, 1) \)[/tex]:
[tex]\[ (-4 + 7, 1 - 6) = (3, -5) \][/tex]
For [tex]\( T' (-4, -3) \)[/tex]:
[tex]\[ (-4 + 7, -3 - 6) = (3, -9) \][/tex]
For [tex]\( U' (-8, -3) \)[/tex]:
[tex]\[ (-8 + 7, -3 - 6) = (-1, -9) \][/tex]
Now, we check to see which translated points fall on a side of square RSTU. Square sides include:
- [tex]\( (-1, -5) \)[/tex] to [tex]\( (3, -5) \)[/tex]
- [tex]\( (3, -5) \)[/tex] to [tex]\( (3, -9) \)[/tex]
- [tex]\( (3, -9) \)[/tex] to [tex]\( (-1, -9) \)[/tex]
- [tex]\( (-1, -9) \)[/tex] to [tex]\( (-1, -5) \)[/tex]
Running through our calculations of the pre-image points of the options:
- Point [tex]\( (2, -9) \)[/tex] (translated from [tex]\((-5, -3)\)[/tex]) does not match pre-image square's sides.
- Point [tex]\( (10, -9) \)[/tex] (translated from [tex]\( (3, -3) \)[/tex]) does not match pre-image square's sides.
- Point [tex]\( (6, -12) \)[/tex] (translated from [tex]\((-1, -6)\)[/tex]) does not match pre-image square's sides.
- Point [tex]\( (11, -15) \)[/tex] (translated from [tex]\( (4, -9) \)[/tex]) does not match pre-image square's sides.
Upon examining:
The point [tex]\((3, -3)\)[/tex] in the translated coordinate matches as it lies on the vertical side from [tex]\( (3, -5) \)[/tex] to [tex]\( (3, -9) \)[/tex] in Square RSTU.
So the point [tex]\( (3, -3) \)[/tex] lies on the side of the pre-image square RSTU.
Therefore, the correct answer is: [tex]\( \boxed{(3, -3)} \)[/tex]
