Question

a line segment has endpoints at (-1,4) and (4,1). which reflection will produce an image with endpoints at (-4,1) and (-1,-4)?
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 the 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 the rules to the endpoints

• For the point \((-1,4)\):
• Reflection across \(x -\)axis: \((-1,4)\to(-1,-4)\)
• Reflection across \(y -\)axis: \((-1,4)\to(1,4)\)
• Reflection across \(y = x\): \((-1,4)\to(4,-1)\)
• Reflection across \(y=-x\): \((-1,4)\to(-4,1)\)
• For the point \((4,1)\):
• Reflection across \(x -\)axis: \((4,1)\to(4,-1)\)
• Reflection across \(y -\)axis: \((4,1)\to(-4,1)\)
• Reflection across \(y = x\): \((4,1)\to(1,4)\)
• Reflection across \(y=-x\): \((4,1)\to(-1,-4)\)

Answer:

a reflection of the line segment across the line \(y=-x\)