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Sagot :
To solve this problem step-by-step, we need to determine which point lies on the translated image of the given square [tex]\( RSTU \)[/tex] after a translation that moves point [tex]\( S \)[/tex] from its original position [tex]\((-4, 1)\)[/tex] to the position [tex]\((3, -5)\)[/tex].
1. Identify the translation vector:
- Original coordinates of point [tex]\( S \)[/tex]: [tex]\((-4, 1)\)[/tex]
- Translated coordinates of point [tex]\( S \)[/tex]: [tex]\((3, -5)\)[/tex]
- Translation vector [tex]\( \vec{v} = (dx, dy) \)[/tex]:
[tex]\[ dx = 3 - (-4) = 7 \quad \text{and} \quad dy = -5 - 1 = -6 \][/tex]
2. Translate all vertices of [tex]\( RSTU \)[/tex] using the translation vector [tex]\( \vec{v} = (7, -6) \)[/tex]:
- Vertex [tex]\( R \)[/tex] at [tex]\((-8, 1)\)[/tex]:
[tex]\[ R' = (-8 + 7, 1 - 6) = (-1, -5) \][/tex]
- Vertex [tex]\( S \)[/tex] at [tex]\((-4, 1)\)[/tex]:
[tex]\[ S' = (-4 + 7, 1 - 6) = (3, -5) \][/tex]
- Vertex [tex]\( T \)[/tex] at [tex]\((-4, -3)\)[/tex]:
[tex]\[ T' = (-4 + 7, -3 - 6) = (3, -9) \][/tex]
- Vertex [tex]\( U \)[/tex] at [tex]\((-8, -3)\)[/tex]:
[tex]\[ U' = (-8 + 7, -3 - 6) = (-1, -9) \][/tex]
3. Check which given option lies on the side of the translated image [tex]\( R'S'T'U' \)[/tex]:
- Square [tex]\( R'S'T'U' \)[/tex] has vertices: [tex]\( R'(-1, -5) \)[/tex], [tex]\( S'(3, -5) \)[/tex], [tex]\( T'(3, -9) \)[/tex], [tex]\( U'(-1, -9) \)[/tex].
- The options are:
- [tex]\((-5, -3)\)[/tex]
- [tex]\((3, -3)\)[/tex]
- [tex]\((-1, -6)\)[/tex]
- [tex]\((4, -9)\)[/tex]
4. Identify the side vertices:
- Side [tex]\( R'S' \)[/tex] from [tex]\((-1, -5)\)[/tex] to [tex]\((3, -5)\)[/tex]
- Side [tex]\( S'T' \)[/tex] from [tex]\((3, -5)\)[/tex] to [tex]\((3, -9)\)[/tex]
- Side [tex]\( T'U' \)[/tex] from [tex]\((3, -9)\)[/tex] to [tex]\((-1, -9)\)[/tex]
- Side [tex]\( U'R' \)[/tex] from [tex]\((-1, -9)\)[/tex] to [tex]\((-1, -5)\)[/tex]
5. Check each option against the sides of [tex]\( R'S'T'U' \)[/tex]:
- [tex]\((-5, -3)\)[/tex]: Not on any side.
- [tex]\((3, -3)\)[/tex]: Not on any side.
- [tex]\((-1, -6)\)[/tex]: This could lie on [tex]\( R'U' \)[/tex] if [tex]\( -1 \)[/tex] matches the x-coordinate and [tex]\(-6 \)[/tex] falls between the y-coordinates [tex]\(-9\)[/tex] and [tex]\(-5\)[/tex], but it doesn’t match.
- [tex]\((4, -9)\)[/tex]: On any side as it does not match the coordinates of any side’s end points.
Since none of the given points lies on the sides of the translated image [tex]\( R'S'T'U' \)[/tex], then the final answer is indeed:
[tex]\[ \text{None of the options given lies on the sides of the translated square.} \][/tex]
1. Identify the translation vector:
- Original coordinates of point [tex]\( S \)[/tex]: [tex]\((-4, 1)\)[/tex]
- Translated coordinates of point [tex]\( S \)[/tex]: [tex]\((3, -5)\)[/tex]
- Translation vector [tex]\( \vec{v} = (dx, dy) \)[/tex]:
[tex]\[ dx = 3 - (-4) = 7 \quad \text{and} \quad dy = -5 - 1 = -6 \][/tex]
2. Translate all vertices of [tex]\( RSTU \)[/tex] using the translation vector [tex]\( \vec{v} = (7, -6) \)[/tex]:
- Vertex [tex]\( R \)[/tex] at [tex]\((-8, 1)\)[/tex]:
[tex]\[ R' = (-8 + 7, 1 - 6) = (-1, -5) \][/tex]
- Vertex [tex]\( S \)[/tex] at [tex]\((-4, 1)\)[/tex]:
[tex]\[ S' = (-4 + 7, 1 - 6) = (3, -5) \][/tex]
- Vertex [tex]\( T \)[/tex] at [tex]\((-4, -3)\)[/tex]:
[tex]\[ T' = (-4 + 7, -3 - 6) = (3, -9) \][/tex]
- Vertex [tex]\( U \)[/tex] at [tex]\((-8, -3)\)[/tex]:
[tex]\[ U' = (-8 + 7, -3 - 6) = (-1, -9) \][/tex]
3. Check which given option lies on the side of the translated image [tex]\( R'S'T'U' \)[/tex]:
- Square [tex]\( R'S'T'U' \)[/tex] has vertices: [tex]\( R'(-1, -5) \)[/tex], [tex]\( S'(3, -5) \)[/tex], [tex]\( T'(3, -9) \)[/tex], [tex]\( U'(-1, -9) \)[/tex].
- The options are:
- [tex]\((-5, -3)\)[/tex]
- [tex]\((3, -3)\)[/tex]
- [tex]\((-1, -6)\)[/tex]
- [tex]\((4, -9)\)[/tex]
4. Identify the side vertices:
- Side [tex]\( R'S' \)[/tex] from [tex]\((-1, -5)\)[/tex] to [tex]\((3, -5)\)[/tex]
- Side [tex]\( S'T' \)[/tex] from [tex]\((3, -5)\)[/tex] to [tex]\((3, -9)\)[/tex]
- Side [tex]\( T'U' \)[/tex] from [tex]\((3, -9)\)[/tex] to [tex]\((-1, -9)\)[/tex]
- Side [tex]\( U'R' \)[/tex] from [tex]\((-1, -9)\)[/tex] to [tex]\((-1, -5)\)[/tex]
5. Check each option against the sides of [tex]\( R'S'T'U' \)[/tex]:
- [tex]\((-5, -3)\)[/tex]: Not on any side.
- [tex]\((3, -3)\)[/tex]: Not on any side.
- [tex]\((-1, -6)\)[/tex]: This could lie on [tex]\( R'U' \)[/tex] if [tex]\( -1 \)[/tex] matches the x-coordinate and [tex]\(-6 \)[/tex] falls between the y-coordinates [tex]\(-9\)[/tex] and [tex]\(-5\)[/tex], but it doesn’t match.
- [tex]\((4, -9)\)[/tex]: On any side as it does not match the coordinates of any side’s end points.
Since none of the given points lies on the sides of the translated image [tex]\( R'S'T'U' \)[/tex], then the final answer is indeed:
[tex]\[ \text{None of the options given lies on the sides of the translated square.} \][/tex]
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