Question

What set of reflections would carry rectangle ABCD onto itself?

Answer 1

The set of reflections that would carry rectangle ABCD back to itself is:
y-axis, x-axis, y-axis, x-axis.

By reflecting the original image over y-axis, the transformed image moves to the 1st quadrant of the cartesian plane. By reflecting the image in quadtrant 1 over x-axis, the image moves to the 4th quadrant, and by reflecting it again over y-axis and another over the x-axis, it goes back to its original position in the 2nd quadrant.

Answer 2

• Reflection about the y-axis has rule (x,y)→(-x,y);
• Reflection about the x-axis has rule: (x,y)→(x,-y).

Consider rectangle ABCD and select, for example point A(-5,1).The set of reflections about the y-axis, x-axis, y-axis, x-axis will act on A in following way:

1. about y-axis A(-5,1)→A'(5,1);
2. about x-axis A'(5,1)→A''(5,-1);
3. about y-axis A''(5,-1)→A'''(-5,-1);
4. about x-axis A'''(-5,-1)→A(-5,1).

This reflections act on all rectangle vertices in the same way, then

the set of reflections that carry rectangle ABCD onto itself is

y-axis, x-axis, y-axis, x-axis