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 reflection rules

The rule for reflecting a point $(x,y)$ across the line $y=-x$ is $(x,y)\to(-y,-x)$.

Step2: Apply rule to first - endpoint

For the point $(-1,4)$, when reflected across $y = -x$, we have $x=-1,y = 4$. Then $(-y,-x)=(-4,1)$.

Step3: Apply rule to second - endpoint

For the point $(4,1)$, when reflected across $y=-x$, we have $x = 4,y=1$. Then $(-y,-x)=(-1,-4)$. So a reflection across the line $y=-x$ produces the desired image endpoints.

Answer:

D. a reflection of the line segment across the line $y = -x$