Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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]
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]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
