A line segment has endpoints at \((3,2)\) and \((2,-3)\). Which reflection will produce an image with endpoints at \((3,-2)\) and \((2,3)\)?

A. A reflection of the line segment across the \(x\)-axis

B. A reflection of the line segment across the \(y\)-axis

C. A reflection of the line segment across the line \(y=x\)

D. A reflection of the line segment across the line [tex]\(y=-x\)[/tex]



Answer :

Let's analyze the given problem step-by-step.

First, we are given the original endpoints of a line segment:
[tex]\[ (3, 2) \quad \text{and} \quad (2, -3) \][/tex]

We are also told the resultant endpoints of the line segment after reflection:
[tex]\[ (3, -2) \quad \text{and} \quad (2, 3) \][/tex]

We need to determine which type of reflection will produce this image. We have four options to consider:

1. Reflection across the x-axis
2. Reflection across the y-axis
3. Reflection across the line \( y = x \)
4. Reflection across the line \( y = -x \)

Reflection across the x-axis:
The transformation rule for reflecting a point \((x, y)\) across the x-axis is:
[tex]\[ (x, y) \rightarrow (x, -y) \][/tex]
Applying this rule to our original points:
[tex]\[ (3, 2) \rightarrow (3, -2) \quad \text{and} \quad (2, -3) \rightarrow (2, 3) \][/tex]
This gives us the points:
[tex]\[ (3, -2) \quad \text{and} \quad (2, 3) \][/tex]
These points match the resultant endpoints perfectly. Therefore, the reflection across the x-axis is the correct transformation.

Reflection across the y-axis:
The transformation rule for reflecting a point \((x, y)\) across the y-axis is:
[tex]\[ (x, y) \rightarrow (-x, y) \][/tex]
Applying this rule to our original points:
[tex]\[ (3, 2) \rightarrow (-3, 2) \quad \text{and} \quad (2, -3) \rightarrow (-2, -3) \][/tex]
This gives us the points:
[tex]\[ (-3, 2) \quad \text{and} \quad (-2, -3) \][/tex]
These points do not match the resultant endpoints.

Reflection across the line \( y = x \):
The transformation rule for reflecting a point \((x, y)\) across the line \( y = x \) is:
[tex]\[ (x, y) \rightarrow (y, x) \][/tex]
Applying this rule to our original points:
[tex]\[ (3, 2) \rightarrow (2, 3) \quad \text{and} \quad (2, -3) \rightarrow (-3, 2) \][/tex]
This gives us the points:
[tex]\[ (2, 3) \quad \text{and} \quad (-3, 2) \][/tex]
These points do not match the resultant endpoints.

Reflection across the line \( y = -x \):
The transformation rule for reflecting a point \((x, y)\) across the line \( y = -x \) is:
[tex]\[ (x, y) \rightarrow (-y, -x) \][/tex]
Applying this rule to our original points:
[tex]\[ (3, 2) \rightarrow (-2, -3) \quad \text{and} \quad (2, -3) \rightarrow (3, -2) \][/tex]
This gives us the points:
[tex]\[ (-2, -3) \quad \text{and} \quad (3, -2) \][/tex]
These points do not match the resultant endpoints.

Therefore, the correct reflection that produces the image with endpoints \((3, -2)\) and \((2, 3)\) is the reflection across the x-axis. The answer is:

[tex]\[ \boxed{\text{a reflection of the line segment across the } x\text{-axis}} \][/tex]