Select the correct answer.
$\triangle A B C$ with vertices $A(-3,0), B(-2,3), C(-1,1)$ is rotated $180^{\circ}$ clockwise about the origin. It is then reflected across the line $y=-x$. What are the coordinates of the vertices of the image?
A. $A^{\prime}(0,3), B^{\prime}(2,3), C(1,1)$
B. $A^{\prime}(0,-3), B^{\prime}(3,-2), C^{\prime}(1,-1)$
C. $A^{\prime}(-3,0), B^{\prime}(-3,2), C(-1,1)$
D. $\left.A^{\prime}(0,-3), B^{\prime}(-2,-3), C^{\prime}-1,-1\right)$
Answer (2)
The problem involves a 18 0 ∘ clockwise rotation about the origin followed by a reflection across the line y = − x .
First, a point ( x , y ) is rotated to ( − x , − y ) .
Then, the rotated point is reflected to ( y , x ) .
Applying this to A ( − 3 , 0 ) , B ( − 2 , 3 ) , and C ( − 1 , 1 ) yields A ′ ( 0 , − 3 ) , B ′ ( 3 , − 2 ) , and C ′ ( 1 , − 1 ) .
The final coordinates are A ′ ( 0 , − 3 ) , B ′ ( 3 , − 2 ) , C ′ ( 1 , − 1 ) .
Explanation
Problem Analysis The problem requires us to find the coordinates of the vertices of a triangle after a 18 0 ∘ clockwise rotation about the origin, followed by a reflection across the line y = − x . We will apply these transformations step by step to each vertex of the triangle.
Transformation Rules A 18 0 ∘ clockwise rotation about the origin transforms a point ( x , y ) to ( − x , − y ) . A reflection across the line y = − x transforms a point ( x , y ) to ( − y , − x ) . Combining these transformations, a point ( x , y ) first becomes ( − x , − y ) after rotation, and then ( − ( − y ) , − ( − x )) = ( y , x ) after reflection. Thus, the combined transformation is ( x , y ) → ( − x , − y ) → ( y , x ) .
Applying Transformations Let's apply the combined transformation to each vertex:
For vertex A ( − 3 , 0 ) , the transformed coordinates are ( 0 , − 3 ) .
For vertex B ( − 2 , 3 ) , the transformed coordinates are ( 3 , − 2 ) .
For vertex C ( − 1 , 1 ) , the transformed coordinates are ( 1 , − 1 ) .
Final Answer The coordinates of the vertices of the image are A ′ ( 0 , − 3 ) , B ′ ( 3 , − 2 ) , and C ′ ( 1 , − 1 ) . Comparing these with the given options, we find that option B matches our result.
Examples
Understanding transformations like rotations and reflections is crucial in computer graphics and game development. For instance, when designing a game, developers use these transformations to animate objects, change perspectives, and create realistic movements. By applying a series of rotations and reflections, a simple object can be manipulated to create complex animations, making the game more engaging and visually appealing. These transformations are also fundamental in creating special effects and simulating physics.
After rotating the triangle's vertices by 180 degrees clockwise about the origin and reflecting them across the line y = -x, the new coordinates are A''(0, -3), B''(3, -2), C''(1, -1). This matches option B. Thus, the final answer is B.
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