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

Step2: Apply rules to endpoints

• For the point \((3,2)\):
• After reflection across \(x\) - axis: \((3,2)\to(3,- 2)\)
• After reflection across \(y\) - axis: \((3,2)\to(-3,2)\)
• After reflection across \(y = x\): \((3,2)\to(2,3)\)
• After reflection across \(y=-x\): \((3,2)\to(-2,-3)\)
• For the point \((2,-3)\):
• After reflection across \(x\) - axis: \((2,-3)\to(2,3)\)
• After reflection across \(y\) - axis: \((2,-3)\to(-2,-3)\)
• After reflection across \(y = x\): \((2,-3)\to(-3,2)\)
• After reflection across \(y=-x\): \((2,-3)\to(3,-2)\)

Since \((3,2)\to(3,-2)\) and \((2,-3)\to(2,3)\) follows the rule \((x,y)\to(x, - y)\) (reflection across the \(x\) - axis), the answer is a reflection of the line segment across the \(x\) - axis.

Answer:

A. a reflection of the line segment across the x - axis