Answered

Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.

Triangle [tex]\(XYZ\)[/tex] has vertices [tex]\(X(0,0)\)[/tex], [tex]\(Y(0,-2)\)[/tex], and [tex]\(Z(-2, 2)\)[/tex]. It is rotated to create the image triangle [tex]\(X'(0,0)\)[/tex], [tex]\(Y'(2,0)\)[/tex], and [tex]\(Z'(2,-2)\)[/tex].

Which rules could describe the rotation? Select two options:

A. [tex]\(R_{0,90^{\circ}}\)[/tex]

B. [tex]\(R_{0,180^{\circ}}\)[/tex]

C. [tex]\(R_{0,270^{\circ}}\)[/tex]

D. [tex]\((x, y) \rightarrow(-y, x)\)[/tex]

E. [tex]\((x, y) \rightarrow(y,-x)\)[/tex]

Sagot :

GinnyAnswer
Given the vertices of triangle [tex]\(XYZ\)[/tex] as [tex]\(X(0,0)\)[/tex], [tex]\(Y(0,-2)\)[/tex], and [tex]\(Z(-2,2)\)[/tex], and the vertices of the image triangle [tex]\(X^{\prime}(0,0)\)[/tex], [tex]\(Y^{\prime}(2,0)\)[/tex], and [tex]\(Z^{\prime}(2,-2)\)[/tex], we need to determine which rotations might transform [tex]\(XYZ\)[/tex] to [tex]\(X^{\prime}Y^{\prime}Z^{\prime}\)[/tex].

We are provided several options:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]
2. [tex]\(R_{0,180^{\circ}}\)[/tex]
3. [tex]\(R_{0,270^{\circ}}\)[/tex]
4. [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
5. [tex]\((x, y) \rightarrow (y, -x)\)[/tex]

Step-by-Step Analysis:

1. Test [tex]\(R_{0, 90^{\circ}}\)[/tex] or [tex]\((x, y) \rightarrow (-y, x)\)[/tex]:
- [tex]\(X(0,0) \rightarrow (-0, 0) = (0, 0)\)[/tex] which matches [tex]\(X^{\prime}(0, 0)\)[/tex].
- [tex]\(Y(0, -2) \rightarrow (2, 0)\)[/tex] which matches [tex]\(Y^{\prime}(2, 0)\)[/tex].
- [tex]\(Z(-2, 2) \rightarrow (-2, -2)\)[/tex], but [tex]\(Z^{\prime} is (2, -2)\)[/tex], so this transformation does not match.

2. Test [tex]\(R_{0, 180^{\circ}}\)[/tex] or [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]:
- [tex]\(X(0,0) \rightarrow (0, 0)\)[/tex] which matches [tex]\(X^{\prime}(0, 0)\)[/tex].
- [tex]\(Y(0, -2) \rightarrow (0, 2)\)[/tex] which does not match [tex]\(Y^{\prime}(2, 0)\)[/tex].
- Therefore, this transformation does not match.

3. Test [tex]\(R_{0, 270^{\circ}}\)[/tex] or [tex]\((x, y) \rightarrow (y, -x)\)[/tex]:
- [tex]\(X(0,0) \rightarrow (0, 0)\)[/tex] which matches [tex]\(X^{\prime}(0, 0)\)[/tex].
- [tex]\(Y(0, -2) \rightarrow (-2, 0)\)[/tex] which does not match [tex]\(Y^{\prime}(2, 0)\)[/tex].
- Therefore, this transformation does not match.

Having checked all possible rotations and their coordinate transformation rules, we see that no combination of the listed rotations can transform triangle [tex]\(XYZ\)[/tex] to triangle [tex]\(X^{\prime}Y^{\prime}Z^{\prime}\)[/tex] correctly.

Thus, the conclusion is:

No transformation among the given options matches the criteria to transform the triangle correctly. Therefore, the answer is:
[tex]\[[]\][/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.