Let's perform the two reflections. First, we reflect PQ in the y-axis.
Next, we will reflect P'Q' in the x-axis.
Now that we have visualized the situation, we can focus on who is correct. First, we will investigate the suggested translation.
Suggested Translation:(x,y) → (x-4,y-5)
Note that if the transformation is a translation, then it will map PQ onto P''Q'' without changing the orientation of the segment. If we perform the suggested translation, however, we notice that P''Q'' has the opposite orientation of PQ.
However, if we rotate PQ by 180^(∘) about the origin, we notice that PQ will map onto P''Q''.
If line k and line m intersect at point P, then a reflection in line k followed by a
reflection in line m is the same as a rotation about point P.
The rotation is 2x^(∘) where x^(∘) is the measure of the acute or right angle formed by the intersecting lines.
In our case, the intersecting lines are the x- and y-axes. They create a 90^(∘) angle between them at the origin. Therefore, the second classmate is correct when he says that a reflection in both axes is the same as a rotation of 180^(∘) about the origin.
