Consider the rules for rotating a figure by 180^(∘) as well as for reflecting a figure in the x- and y-axes.
Our friend is correct.
Practice makes perfect
When you rotate a figure by 180^(∘), the coordinates of the figures vertices will transform in the following way:
(a,b) → (- a, - b)
Assuming that one of the figures vertices is at (x_1,y_1), then the corresponding vertex on the image after a 180^(∘) rotation will be (- x_1, - y_1). Let's illustrate this.
If we can establish that this same transformation will occur after a reflection in the y-axis followed by a reflection in the x-axis, then we know our friend is correct.
Reflection in the y-axis
If (a,b) is reflected in the y-axis, then it's image is the point (- a,b). In other words, the x-coordinate changes sign while the y-coordinate stays the same. Therefore, for the vertex (x_1,y_1), the coordinates of the corresponding vertex on our image after a reflection in the y-axis will be (- x_1,y_1).
Reflection in the x-axis
If (a,b) is reflected in the x-axis, then it's image is the point (a,- b). In other words, the y-coordinate changes sign while the x-coordinate stays the same. Therefore, for the vertex (- x_1,y_1), the coordinates of the corresponding vertex after a reflection in the x-axis will be (- x_1,- y_1).
As we can see, the two reflections result in the same transformation as a rotation of 180^(∘) about the origin. Therefore, our friend is correct.
