Question

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 reflection of the line segment across the x-axis
○ a reflection of the line segment across the y-axis
○ a reflection of the line segment across the line y = x
○ a reflection of the line segment across the line y = -x

Explanation:

Step1: Recall reflection rules

Reflection across x - axis: \((x,y)\to(x, - y)\)
Reflection across y - axis: \((x,y)\to(-x,y)\)
Reflection across \(y = x\): \((x,y)\to(y,x)\)
Reflection across \(y=-x\): \((x,y)\to(-y,-x)\)

Step2: Analyze the given points

Original endpoints: \((3,2)\) and \((2, - 3)\)
Image endpoints: \((3,-2)\) and \((2,3)\)

For the point \((3,2)\), applying reflection across x - axis: \((3,2)\to(3,-2)\)
For the point \((2,-3)\), applying reflection across x - axis: \((2,-3)\to(2,3)\)
Which matches the image points.

Let's check other reflections:

• Reflection across y - axis: \((3,2)\to(-3,2)

eq(3,-2)\), so not y - axis.

• Reflection across \(y = x\): \((3,2)\to(2,3)\) and \((2,-3)\to(-3,2)\), which does not match the image points.
• Reflection across \(y=-x\): \((3,2)\to(-2,-3)

eq(3,-2)\), so not \(y = - x\).

Answer:

a reflection of the line segment across the x - axis