Which graph shows the image of a figure after a rotation of 180° around the origin?
A. graph AThe coordinates of the vertices for the first square are upper A superscript 1 baseline left-parenthesis negative 2 comma 3 right-parenthesis, upper B superscript 1 baseline left-parenthesis negative 3 comma 1 right-parenthesis, upper C superscript 1 baseline left-parenthesis negative 1 comma 0 right-parenthesis, and upper D superscript 1 baseline left-parenthesis 2 comma 0 right-parenthesis. The coordinates of the vertices for the second square are upper A left-parenthesis 2 comma 3 right-parenthesis, upper B left-parenthesis 3 comma 1 right-parenthesis, upper C left-parenthesis 1 comma 0 right-parenthesis, and upper D left-parenthesis 2 comma 0 right-parenthesis.
B. graph BThe coordinates of the vertices for the first square are upper A superscript 1 baseline left-parenthesis negative 2 comma 3 right-parenthesis, upper B superscript 1 baseline left-parenthesis negative 3 comma 1 right-parenthesis, upper C superscript 1 baseline left-parenthesis negative 1 comma 0 right-parenthesis, and upper D superscript 1 baseline left-parenthesis 2 comma 0 right-parenthesis. The coordinates of the vertices for the second square are upper A left-parenthesis negative 2 comma 3 right-parenthesis, upper B left-parenthesis 1 comma negative 1 right-parenthesis, upper C left-parenthesis 1 comma 0 right-parenthesis, and upper D left-parenthesis 0 comma negative 2 right-parenthesis.
C. graph CThe coordinates of the vertices for the first square are upper A superscript 1 baseline left-parenthesis negative 2 comma 3 right-parenthesis, upper B superscript 1 baseline left-parenthesis negative 3 comma 1 right-parenthesis, upper C superscript 1 baseline left-parenthesis negative 1 comma 0 right-parenthesis, and upper D superscript 1 baseline left-parenthesis 2 comma 0 right-parenthesis. The coordinates of the vertices for the second square are upper A left-parenthesis negative 5 comma negative 2 right-parenthesis, upper B left-parenthesis negative 4 comma negative 4 right-parenthesis, upper C left-parenthesis negative 2 comma negative 3 right-parenthesis, and upper D left-parenthesis negative 3 comma negative 1 right-parenthesis.
D. graph DThe coordinates of the vertices for the first square are upper A superscript 1 baseline left-parenthesis negative 2 comma 3 right-parenthesis, upper B superscript 1 baseline left-parenthesis negative 3 comma 1 right-parenthesis, upper C superscript 1 baseline left-parenthesis negative 1 comma 0 right-parenthesis, and upper D superscript 1 baseline left-parenthesis 2 comma 0 right-parenthesis. The coordinates of the vertices for the second square are upper A left-parenthesis negative 2 comma negative 3 right-parenthesis, upper B left-parenthesis 1 comma negative 4 right-parenthesis, upper C left-parenthesis 2 comma negative 1 right-parenthesis, and upper D left-parenthesis negative 1 comma 0 right-parenthesis.
B. graph B
The correct answer is:
B. graph B-The coordinates of the vertices for the first square are A'(−2, 3), B'(-3, 1), C'(-1, 0), and D'(2, 0). The coordinates of the vertices for the second square are A'(-2, 3), B'(1, -1), C'(1, 0), and D'(0, -2).
To find the graph that shows the image of a figure after a rotation of 180° around the origin, we need to analyze the given options.
In a 180° rotation around the origin, each point is reflected across the x-axis and y-axis simultaneously. This means that the x-coordinate of each point becomes its negative, and the y-coordinate also becomes its negative.
Let's analyze the options:
A. graph A:
- The first square has vertices (-2,3), (-3,1), (-1,0), and (2,0).
- The coordinates of the vertices for the second square are (2,3), (3,1), (1,0), and (2,0).
- This option does not show the correct reflection across both x-axis and y-axis.
B. graph B:
- The first square has vertices (-2,3), (-3,1), (-1,0), and (2,0).
- The coordinates of the vertices for the second square are (-2,3), (1,-1), (1,0), and (0,-2).
- This option does show the correct reflection across both x-axis and y-axis.
C. graph C:
- The first square has vertices (-2,3), (-3,1), (-1,0), and (2,0).
- The coordinates of the vertices for the second square are (-5,-2), (-4,-4), (-2,-3), and (-3,-1).
- This option does not show the correct reflection across both x-axis and y-axis.
D. graph D:
- The first square has vertices (-2,3), (-3,1), (-1,0), and (2,0).
- The coordinates of the vertices for the second square are (-2,-3), (1,-4), (2,-1), and (-1,0).
- This option does not show the correct reflection across both x-axis and y-axis.
Therefore, the graph that shows the image of a figure after a rotation of 180° around the origin is graph B.