Question

Which transformation will be equivalent to rotating a figure 180° counterclockwise?

A) reflecting over the line y = x.
B) reflecting over the line y = -x.
C) reflecting over the x-axis and the y-axis.
D) translating left 3 units and down 5 units.

Answer 1

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.

Answer 2

The transformation which is equivalent to rotating a figure $180^{\circ}$ counterclockwise is reflecting over the $x$-axis and $y$-axis. Therefore, the $\fbox{\begin\\\ \bf option (C)\\\end{minispace}}$ is correct.

Further explanation:

Consider a coordinate in the form $(a,b)$ where $a$ and $b$ are real numbers.

If $a$ and $b$ are positive then the point $(a,b)$ lies in the first quadrant.

If we rotate the coordinate $(a,b)$ $180^{\circ}$ counterclockwise then the coordinates become $(-a,-b)$.

The coordinate $(-a,-b)$ lies in the third quadrant.

If we reflect the coordinate $(a,b)$ about the $x$-axis then the coordinates become $(a,-b)$.

The coordinate $(a,-b)$ lies in the fourth quadrant.

If we reflect the coordinate $(a,-b)$ about the $y$-axis then the coordinates become $(-a,-b)$.

The coordinate $(-a,-b)$ lies in the third quadrant.

This implies that rotating a figure $180^{\circ}$ counterclockwise is equivalent to transformation of reflecting over $x$-axis and reflecting over $y$-axis.

Option (A)

In option (A) it is given that reflection about the line $y=x$ is equivalent to rotating a figure $180^{\circ}$ counterclockwise.

If we reflect the coordinate $(a,b)$ about the line $y=x$ then the coordinates become $(b,a)$.

This is not same as rotating the figure $180^{\circ}$ counterclockwise.

Therefore, the option (A) is incorrect.

Option (B)

In option (B) it is given that reflection about $y=-x$ is equivalent to rotating a figure $180^{\circ}$ counterclockwise.

If we reflect the coordinate $(a,b)$ about the line $y=-x$ then the coordinates become $(-b,-a)$.

This is not same as rotating the figure $180^{\circ}$ counterclockwise.

Therefore, the option (B) is incorrect.

Option (C)

In option (C) it is given that reflection about $x$-axis and $y$-axis is equivalent to rotating a figure $180^{\circ}$ counterclockwise.

If we reflect the coordinate $(a,b)$ about $x$-axis and $y$-axis then the coordinates become $(-a,-b)$.

This is same as rotating the figure $180^{\circ}$ counterclockwise.

Therefore, the option (C) is correct.

Option (D)

In option (D) it is given that shifting a point $3$ units left and $5$ units down is equivalent to rotating a figure $180^{\circ}$ counterclockwise.

If we shift the point $(a,b)$, $3$ units left and $5$ units down the coordinate is $(a+3,b-5)$.

This is not same as rotating the figure $180^{\circ}$ counterclockwise.

Therefore, the option (D) is incorrect.

Therefore, the $\fbox{\begin\\\ \bf option (C)\\\end{minispace}}$ 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.