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)?
o 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: Check each option

• Option 1: Reflection across \(x -\)axis
• For \((-1,4)\): \((-1,4)\to(-1,-4)\)
• For \((4,1)\): \((4,1)\to(4,-1)

eq(-4,1)\)

• Option 2: Reflection across \(y -\)axis
• For \((-1,4)\): \((-1,4)\to(1,4)

eq(-1,-4)\)

• For \((4,1)\): \((4,1)\to(-4,1)\)
• Option 3: Reflection across \(y = x\)
• For \((-1,4)\): \((-1,4)\to(4,-1)

eq(-1,-4)\)

• For \((4,1)\): \((4,1)\to(1,4)

eq(-4,1)\)

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

Answer:

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