A triangle is rotated $90^{\circ}$ about the origin. Which rule describes the transformation?
A. $(x, y) \rightarrow(-x,-y)$
B. $(x, y) \rightarrow(-y, x)$
C. $(x, y) \rightarrow(-y,-x)$
D. $(x, y) \rightarrow(y,-x)$
Answer (1)
A 9 0 ∘ rotation about the origin transforms a point ( x , y ) to ( − y , x ) .
This transformation can be verified using example points like ( 1 , 0 ) and ( 0 , 1 ) .
The rule ( x , y ) → ( − y , x ) accurately describes this rotation.
The final answer is ( x , y ) → ( − y , x ) .
Explanation
Problem Analysis Let's analyze the effect of a 9 0 ∘ rotation about the origin on a point ( x , y ) in the coordinate plane. We need to determine which of the given rules correctly describes this transformation.
Rotation Transformation Consider a point ( x , y ) . After a 9 0 ∘ counterclockwise rotation about the origin, the new coordinates become ( − y , x ) . This can be derived using rotation matrices or by considering the geometric transformation.
Verification with Example Points Let's test this with a simple point, say ( 1 , 0 ) . After a 9 0 ∘ rotation, it becomes ( 0 , 1 ) . This matches the rule ( x , y ) → ( − y , x ) .
Let's test another point, say ( 0 , 1 ) . After a 9 0 ∘ rotation, it becomes ( − 1 , 0 ) . This also matches the rule ( x , y ) → ( − y , x ) .
Identifying the Correct Rule Comparing our result with the given options, we find that the correct rule is ( x , y ) → ( − y , x ) .
Final Answer Therefore, the rule that describes the transformation of a 9 0 ∘ rotation about the origin is ( x , y ) → ( − y , x ) .
Examples
Imagine you're designing a video game where objects need to rotate around a central point. Understanding coordinate transformations, like a 90-degree rotation, is crucial for accurately updating the positions of these objects on the screen. For example, if a character at position (2, 3) needs to rotate 90 degrees around the origin, you'd apply the rule (x, y) -> (-y, x) to find the new position (-3, 2). This ensures smooth and realistic movement in your game.
