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. 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
Answer (2)
A reflection across the x-axis transforms a point ( x , y ) to ( x , − y ) .
Applying this transformation to the point ( 3 , 2 ) results in ( 3 , − 2 ) , and applying it to the point ( 2 , − 3 ) results in ( 2 , 3 ) .
Since the transformed points match the desired image endpoints, the correct reflection is across the x-axis.
Therefore, the answer is a reflection across the x -axis: a reflection of the line segment across the x -axis .
Explanation
Analyze the effect of each reflection. Let's analyze the effect of each reflection on the given endpoints ( 3 , 2 ) and ( 2 , − 3 ) .
Reflection across the x-axis: A point ( x , y ) reflected across the x-axis becomes ( x , − y ) .
( 3 , 2 ) becomes ( 3 , − 2 ) .
( 2 , − 3 ) becomes ( 2 , − ( − 3 )) = ( 2 , 3 ) .
This matches the desired image endpoints ( 3 , − 2 ) and ( 2 , 3 ) .
Reflection across the y-axis: A point ( x , y ) reflected across the y-axis becomes ( − x , y ) .
( 3 , 2 ) becomes ( − 3 , 2 ) .
( 2 , − 3 ) becomes ( − 2 , − 3 ) .
This does not match the desired image endpoints.
Reflection across the line y = x : A point ( x , y ) reflected across the line y = x becomes ( y , x ) .
( 3 , 2 ) becomes ( 2 , 3 ) .
( 2 , − 3 ) becomes ( − 3 , 2 ) .
This does not match the desired image endpoints.
Reflection across the line y = − x : A point ( x , y ) reflected across the line y = − x becomes ( − y , − x ) .
( 3 , 2 ) becomes ( − 2 , − 3 ) .
( 2 , − 3 ) becomes ( − ( − 3 ) , − 2 ) = ( 3 , − 2 ) .
This does not match the desired image endpoints.
Identify the correct reflection. The reflection across the x-axis maps the original endpoints ( 3 , 2 ) and ( 2 , − 3 ) to the image endpoints ( 3 , − 2 ) and ( 2 , 3 ) .
Conclusion. Therefore, the reflection that produces the image with endpoints at ( 3 , − 2 ) and ( 2 , 3 ) is a reflection across the x-axis.
Examples
Reflections are used in various real-world applications, such as creating symmetrical designs in architecture, understanding how light behaves in mirrors, and in computer graphics to generate mirror images or symmetrical patterns. For instance, when designing a building, architects use reflections to ensure that the left and right sides are balanced and aesthetically pleasing. Similarly, in physics, understanding reflections is crucial for designing optical instruments like telescopes and microscopes.
The reflection that transforms the endpoints from ( 3 , 2 ) and ( 2 , − 3 ) to ( 3 , − 2 ) and ( 2 , 3 ) is a reflection across the x-axis. Hence, the correct option is A.
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