Answered

Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

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]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.