Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

If triangle [tex]$XYZ$[/tex] is translated using the rule [tex]\((x, y) \rightarrow (x+4, y-1)\)[/tex] and then reflected across the [tex]\(y\)[/tex]-axis to create triangle [tex]\(X'Y'Z'\)[/tex], what is the location of [tex]\(X'\)[/tex]?

A. [tex]\((8, 4)\)[/tex]

B. [tex]\((-5, 2)\)[/tex]

C. [tex]\((-8, 4)\)[/tex]

D. [tex]\((-9, 2)\)[/tex]


Sagot :

To find the location of [tex]\( X^ \)[/tex] starting with the point [tex]\( X \)[/tex] at coordinates [tex]\((8, 4)\)[/tex], we need to follow two transformations: a translation and a reflection. Here are the steps:

1. Translation: The translation rule given is [tex]\((x, y) \rightarrow (x + 4, y - 1)\)[/tex].

- For the point [tex]\( X \)[/tex] at coordinates [tex]\((8, 4)\)[/tex]:
- The new [tex]\( x \)[/tex]-coordinate after translation: [tex]\( 8 + 4 = 12 \)[/tex].
- The new [tex]\( y \)[/tex]-coordinate after translation: [tex]\( 4 - 1 = 3 \)[/tex].
- So, after translation, the coordinates are [tex]\((12, 3)\)[/tex].

2. Reflection across the [tex]\( y \)[/tex]-axis: Reflecting a point across the [tex]\( y \)[/tex]-axis changes the sign of the [tex]\( x \)[/tex]-coordinate while keeping the [tex]\( y \)[/tex]-coordinate the same.

- For the translated point at coordinates [tex]\((12, 3)\)[/tex]:
- The new [tex]\( x \)[/tex]-coordinate after reflection: [tex]\(-12\)[/tex].
- The [tex]\( y \)[/tex]-coordinate remains the same: [tex]\( 3 \)[/tex].
- So, after reflection, the coordinates are [tex]\((-12, 3)\)[/tex].

Hence, the location of [tex]\( X^
\)[/tex] after performing both transformations is [tex]\((-12, 3)\)[/tex].
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.