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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 :

GinnyAnsweridn
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.
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