The image of the point $(4,-2)$ under a rotation 180 degrees about the origin is:
A. $(-4,-2)$
B. $(-2,4)$
C. $(-2,-4)$
D. $(-4,2)$
Answer (2)
A 180-degree rotation about the origin transforms a point ( x , y ) to ( − x , − y ) .
Apply the transformation to the point ( 4 , − 2 ) .
Calculate the new coordinates: x ′ = − 4 , y ′ = − ( − 2 ) = 2 .
The image of the point ( 4 , − 2 ) is ( − 4 , 2 ) .
Explanation
Understanding 180-degree Rotation When a point ( x , y ) is rotated 180 degrees about the origin, its image is the point ( − x , − y ) . This is because a 180-degree rotation changes the signs of both the x-coordinate and the y-coordinate.
Applying the Rotation Rule Given the point ( 4 , − 2 ) , we apply the 180-degree rotation rule:
x ′ = − x = − 4 y ′ = − y = − ( − 2 ) = 2
So, the image of the point ( 4 , − 2 ) after a 180-degree rotation about the origin is ( − 4 , 2 ) .
Final Answer Therefore, the image of the point ( 4 , − 2 ) under a rotation of 180 degrees about the origin is ( − 4 , 2 ) .
Examples
Imagine you're designing a symmetrical pattern around a central point. If you have a design element at coordinates (4, -2) and you want to create a symmetrical element on the opposite side, you would rotate the original point 180 degrees about the center (origin). This rotation would place the new element at (-4, 2), ensuring perfect symmetry in your design. This principle is used in various fields, including graphic design, architecture, and even physics, to create balanced and aesthetically pleasing structures and patterns.
The image of the point ( 4 , − 2 ) after a 180-degree rotation about the origin is ( − 4 , 2 ) . Therefore, the correct answer is option D: ( − 4 , 2 ) .
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