Answered

Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

Which reflection will produce an image of [tex]$\triangle RST$[/tex] with a vertex at [tex]$(2, -3)$[/tex]?

A. A reflection of [tex]$\triangle RST$[/tex] across the [tex][tex]$x$[/tex][/tex]-axis
B. A reflection of [tex]$\triangle RST$[/tex] across the [tex]$y$[/tex]-axis
C. A reflection of [tex]$\triangle RST$[/tex] across the line [tex][tex]$y = x$[/tex][/tex]
D. A reflection of [tex]$\triangle RST$[/tex] across the line [tex]$y = -x$[/tex]

Sagot :

GinnyAnswer
To determine the new coordinates of the vertex after the reflections, let's go through each type of reflection step-by-step:

1. Reflection across the [tex]\( x \)[/tex]-axis:

When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\( x \)[/tex]-axis, its new coordinates are [tex]\((x, -y)\)[/tex].

For the vertex [tex]\( (2, -3) \)[/tex]:
[tex]\[ (2, -(-3)) = (2, 3) \][/tex]

2. Reflection across the [tex]\( y \)[/tex]-axis:

When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\( y \)[/tex]-axis, its new coordinates are [tex]\((-x, y)\)[/tex].

For the vertex [tex]\( (2, -3) \)[/tex]:
[tex]\[ (-2, -3) \][/tex]

3. Reflection across the line [tex]\( y = x \)[/tex]:

When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\( y = x \)[/tex], its new coordinates are [tex]\((y, x)\)[/tex].

For the vertex [tex]\( (2, -3) \)[/tex]:
[tex]\[ (-3, 2) \][/tex]

4. Reflection across the line [tex]\( y = -x \)[/tex]:

When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\( y = -x \)[/tex], its new coordinates are [tex]\((-y, -x)\)[/tex].

For the vertex [tex]\( (2, -3) \)[/tex]:
[tex]\[ (-(-3), -2) = (3, -2) \][/tex]

Now, if we compare these results:

- The reflection across the [tex]\( x \)[/tex]-axis gives us [tex]\((2, 3)\)[/tex]
- The reflection across the [tex]\( y \)[/tex]-axis gives us [tex]\((-2, -3)\)[/tex]
- The reflection across the line [tex]\( y = x \)[/tex] gives us [tex]\((-3, 2)\)[/tex]
- The reflection across the line [tex]\( y = -x \)[/tex] gives us [tex]\((3, -2)\)[/tex]

Therefore, the new positions of the vertex [tex]\( (2, -3) \)[/tex] after the reflections are:

- Reflecting across the [tex]\( x \)[/tex]-axis will produce an image with the vertex at [tex]\( (2, 3) \)[/tex]
- Reflecting across the [tex]\( y \)[/tex]-axis will produce an image with the vertex at [tex]\( (-2, -3) \)[/tex]
- Reflecting across the line [tex]\( y = x \)[/tex] will produce an image with the vertex at [tex]\( (-3, 2) \)[/tex]
- Reflecting across the line [tex]\( y = -x \)[/tex] will produce an image with the vertex at [tex]\( (3, -2) \)[/tex]

These corresponding reflections provide the correct new positions for the vertex after each type of reflection.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.