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. A reflection of the line segment across the $x$-axis
B. A reflection of the line segment across the $y$-axis
C. A reflection of the line segment across the line $y=x$
D. A reflection of the line segment across the line $y=-x

The correct reflection that transforms the line segment with endpoints at ( − 1 , 4 ) and ( 4 , 1 ) to new endpoints at ( − 4 , − 1 ) and ( − 1 , − 4 ) is across the line y = − x . Therefore, the answer is option D. This transformation correctly maps the points as required.
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Answered by Anonymous | 2025-07-04

Check reflection across the x-axis: ( x , y ) → ( x , − y ) .
Check reflection across the y-axis: ( x , y ) → ( − x , y ) .
Check reflection across the line y = x : ( x , y ) → ( y , x ) .
Check reflection across the line y = − x : ( x , y ) → ( − y , − x ) . The correct transformation is reflection across the line y = − x , which maps ( − 1 , 4 ) to ( − 4 , 1 ) and ( 4 , 1 ) to ( − 1 , − 4 ) .
reflection across the line y = − x ​

Explanation

Problem Analysis Let's analyze the problem. We have a line segment with endpoints A(-1, 4) and B(4, 1). We want to find the reflection that transforms this line segment to a new line segment with endpoints A'(-4, 1) and B'(-1, -4). We will consider each of the given reflection options and see which one maps A to A' and B to B'.

Checking each reflection option

Reflection across the x-axis: This transformation maps (x, y) to (x, -y).

A(-1, 4) becomes (-1, -4)
B(4, 1) becomes (4, -1) This is not the desired transformation.


Reflection across the y-axis: This transformation maps (x, y) to (-x, y).

A(-1, 4) becomes (1, 4)
B(4, 1) becomes (-4, 1) This is not the desired transformation.


Reflection across the line y = x: This transformation maps (x, y) to (y, x).

A(-1, 4) becomes (4, -1)
B(4, 1) becomes (1, 4) This is not the desired transformation.


Reflection across the line y = -x: This transformation maps (x, y) to (-y, -x).

A(-1, 4) becomes (-4, -(-1)) = (-4, 1)
B(4, 1) becomes (-1, -4) This is the desired transformation.


Final Answer Therefore, the reflection that produces the image with endpoints at (-4, 1) and (-1, -4) is the reflection across the line y = -x.


Examples
Reflections are used in computer graphics to create symmetrical images or mirror effects. For example, reflecting an object across the y-axis can create a mirrored duplicate on the other side of the screen. This is commonly used in games and design applications to efficiently create symmetrical designs or environments.

Answered by GinnyAnswer | 2025-07-04