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Square RSTU is translated to form R'S'T'U', which has vertices [tex]\(R^{\prime}(-8,1), S^{\prime}(-4,1), T^{\prime}(-4,-3),\)[/tex] and [tex]\(U^{\prime}(-8,-3)\)[/tex]. If point [tex]\(S\)[/tex] has coordinates of [tex]\((3,-5)\)[/tex], which point lies on a side of the pre-image, square RSTU?

A. [tex]\((-5,-3)\)[/tex]
B. [tex]\((3,-3)\)[/tex]
C. [tex]\((-1,-6)\)[/tex]
D. [tex]\((4,-9)\)[/tex]

Sagot :

GinnyAnswer
To solve the problem, we need to determine which of the given points lies on a side of the pre-image square RSTU.

Here are the steps:

1. Identify the Translation Vector:
Calculate the translation vector used to translate point [tex]\( S \)[/tex] with coordinates [tex]\( (3, -5) \)[/tex] to [tex]\( S' \)[/tex] with coordinates [tex]\( (-4, 1) \)[/tex].

Translation vector [tex]\( \vec{v} \)[/tex] can be found by subtracting the coordinates of [tex]\( S \)[/tex] from [tex]\( S' \)[/tex]:
[tex]\[ \vec{v} = (-4 - 3, 1 - (-5)) \][/tex]
[tex]\[ \vec{v} = (-4 - 3, 1 + 5) \][/tex]
[tex]\[ \vec{v} = (-7, 6) \][/tex]

2. Determine the Pre-Image Points:
Translate each of the given points of the square R'S'T'U' back using the inverse of the translation vector, [tex]\( -\vec{v} \)[/tex].

- For [tex]\( R'(-8, 1) \)[/tex]:
[tex]\[ R = (-8 + 7, 1 - 6) = (-1, -5) \][/tex]
- For [tex]\( S'(-4, 1) \)[/tex]:
[tex]\[ S = (-4 + 7, 1 - 6) = (3, -5) \][/tex]
- For [tex]\( T'(-4, -3) \)[/tex]:
[tex]\[ T = (-4 + 7, -3 - 6) = (3, -9) \][/tex]
- For [tex]\( U'(-8, -3) \)[/tex]:
[tex]\[ U = (-8 + 7, -3 - 6) = (-1, -9) \][/tex]

Therefore, the pre-image points (vertices of square RSTU) are:
[tex]\[ R = (-1, -5), S = (3, -5), T = (3, -9), U = (-1, -9) \][/tex]

3. Identify the Correct Point:
Now, we need to check which of the given points lies on the side of the pre-image square RSTU. The given points are:
[tex]\((-5, -3)\)[/tex], [tex]\((3, -3)\)[/tex], [tex]\((-1, -6)\)[/tex], [tex]\((4, -9)\)[/tex].

Let's examine each point:

- [tex]\((-5, -3)\)[/tex]: This point does not match any side of the square RSTU.
- [tex]\((3, -3)\)[/tex]: This point does not match any side of the square RSTU.
- [tex]\((-1, -6)\)[/tex]: This point does not match any side of the square RSTU.
- [tex]\((4, -9)\)[/tex]: This point does not match any side of the square RSTU.

None of these points correspond to any side of the pre-image square RSTU, and therefore, none of the provided points lie on the side of the pre-image square.

So, the final answer is:
[tex]\[ \boxed{\text{None}} \][/tex]
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