What is the image of point $A(4,2)$ after the composition of transformations defined by $R_{90^{\circ}} \circ r_{y=x}$ ?
1) $(-4,2)$
2) $(4,-2)$
3) $(-4,-2)$
4) $(2,-4)$
Answer (2)
The image of point A ( 4 , 2 ) after reflecting over the line y = x and then rotating 90 degrees counterclockwise is ( − 4 , 2 ) . The transformations were applied step-by-step: first a reflection resulted in ( 2 , 4 ) , and then a rotation produced ( − 4 , 2 ) . The final answer is ( − 4 , 2 ) .
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Reflect the point A ( 4 , 2 ) over the line y = x to get ( 2 , 4 ) .
Rotate the point ( 2 , 4 ) by 9 0 ∘ counterclockwise to get ( − 4 , 2 ) .
The final image of the point A ( 4 , 2 ) after the composition of transformations is ( − 4 , 2 ) .
Explanation
Problem Analysis The problem asks for the image of point A ( 4 , 2 ) after a composition of two transformations: a reflection over the line y = x , followed by a 90-degree counterclockwise rotation about the origin.
Reflection over y=x First, we apply the reflection r y = x to the point A ( 4 , 2 ) . The reflection of a point ( x , y ) over the line y = x is given by ( y , x ) . Therefore, the image of A ( 4 , 2 ) after the reflection is ( 2 , 4 ) .
Rotation by 90 degrees Next, we apply the rotation R 9 0 ∘ to the point ( 2 , 4 ) . The rotation of a point ( x , y ) by 9 0 ∘ counterclockwise is given by ( − y , x ) . Therefore, the image of ( 2 , 4 ) after the rotation is ( − 4 , 2 ) .
Final Answer Therefore, the image of point A ( 4 , 2 ) after the composition of transformations R 9 0 ∘ ∘ r y = x is ( − 4 , 2 ) .
Examples
Understanding transformations is crucial in computer graphics and animation. For instance, when designing a game, you might reflect a character across a line (like y=x) to create a mirrored version or rotate it to change its orientation. These transformations help create realistic and engaging visual experiences for players.
