Respuesta :
If we have given coordinates of the image are in form (h,k).
The resulting coordinates of image rotation of 180° around the origin would be (-h,-k).
We have rule (h,k) ---> (-h,-k).
We can see that x-coordinate is being multiplied by -1 and then y-coordinate is also being multiplied by -1.
Above rule could be break into two parts.
(h,k) ---> (-h,k) ----> (-h,-k).
We can see in first step, (h,k) ---> (-h,k) is being reflecting over the x-axis and
in second step (-h,k) ----> (-h,-k) is being reflecting over the y-axis.
Therefore, correct option is C) reflecting over the x-axis and the y-axis.
The transformation which is equivalent to rotating a figure [tex]180^{\circ}[/tex] counterclockwise is reflecting over the [tex]x[/tex]-axis and [tex]y[/tex]-axis. Therefore, the [tex]\fbox{\begin\\\ \bf option (C)\\\end{minispace}}[/tex] is correct.
Further explanation:
Consider a coordinate in the form [tex](a,b)[/tex] where [tex]a[/tex] and [tex]b[/tex] are real numbers.
If [tex]a[/tex] and [tex]b[/tex] are positive then the point [tex](a,b)[/tex] lies in the first quadrant.
If we rotate the coordinate [tex](a,b)[/tex] [tex]180^{\circ}[/tex] counterclockwise then the coordinates become [tex](-a,-b)[/tex].
The coordinate [tex](-a,-b)[/tex] lies in the third quadrant.
If we reflect the coordinate [tex](a,b)[/tex] about the [tex]x[/tex]-axis then the coordinates become [tex](a,-b)[/tex].
The coordinate [tex](a,-b)[/tex] lies in the fourth quadrant.
If we reflect the coordinate [tex](a,-b)[/tex] about the [tex]y[/tex]-axis then the coordinates become [tex](-a,-b)[/tex].
The coordinate [tex](-a,-b)[/tex] lies in the third quadrant.
This implies that rotating a figure [tex]180^{\circ}[/tex] counterclockwise is equivalent to transformation of reflecting over [tex]x[/tex]-axis and reflecting over [tex]y[/tex]-axis.
Option (A)
In option (A) it is given that reflection about the line [tex]y=x[/tex] is equivalent to rotating a figure [tex]180^{\circ}[/tex] counterclockwise.
If we reflect the coordinate [tex](a,b)[/tex] about the line [tex]y=x[/tex] then the coordinates become [tex](b,a)[/tex].
This is not same as rotating the figure [tex]180^{\circ}[/tex] counterclockwise.
Therefore, the option (A) is incorrect.
Option (B)
In option (B) it is given that reflection about [tex]y=-x[/tex] is equivalent to rotating a figure [tex]180^{\circ}[/tex] counterclockwise.
If we reflect the coordinate [tex](a,b)[/tex] about the line [tex]y=-x[/tex] then the coordinates become [tex](-b,-a)[/tex].
This is not same as rotating the figure [tex]180^{\circ}[/tex] counterclockwise.
Therefore, the option (B) is incorrect.
Option (C)
In option (C) it is given that reflection about [tex]x[/tex]-axis and [tex]y[/tex]-axis is equivalent to rotating a figure [tex]180^{\circ}[/tex] counterclockwise.
If we reflect the coordinate [tex](a,b)[/tex] about [tex]x[/tex]-axis and [tex]y[/tex]-axis then the coordinates become [tex](-a,-b)[/tex].
This is same as rotating the figure [tex]180^{\circ}[/tex] counterclockwise.
Therefore, the option (C) is correct.
Option (D)
In option (D) it is given that shifting a point [tex]3[/tex] units left and [tex]5[/tex] units down is equivalent to rotating a figure [tex]180^{\circ}[/tex] counterclockwise.
If we shift the point [tex](a,b)[/tex], [tex]3[/tex] units left and [tex]5[/tex] units down the coordinate is [tex](a+3,b-5)[/tex].
This is not same as rotating the figure [tex]180^{\circ}[/tex] counterclockwise.
Therefore, the option (D) is incorrect.
Therefore, the [tex]\fbox{\begin\\\ \bf option (C)\\\end{minispace}}[/tex] is correct.
Learn more
1. Learn more about the rotation of the triangle about the origin https://brainly.com/question/7437053.
2. Learn more about when a triangle is rotated about the origin https://brainly.com/question/2992432.
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Geometry
Keywords: Transformation, rotation, reflection, clockwise, geometry, counterclockwise, -axis, - axis, coordinates, graph, origin, line, degrees, translation, symmetry.
