Question

What set of reflections would carry rectangle ABCD onto itself?

A) y-axis, x-axis, y-axis
B) x-axis, y-axis, y-axis
C) x-axis, y=x, x-axis, y=x
D) y=x, x-axis, y=x, y-axis

Answer 1

Answer: option D) y=x, x-axis, y=x, y-axis.

I first thought it was the option C) and I tried with it but it was wrong. This is how I dit it.

Option C step by step:

1) Reflection over the x - axis => point with coordinates (a,b) is transformed into point with coordinates (a, -b)

2) Reflection over the line y = x => point with coordinates (a, -b) is transformed into point with coordinates (-b,a)

3) New feflection over the x - axis => (-b,a) transforms into (-b, -a)

4) New reflection over the line y = x => (-b,-a) transforms into (-a,-b)

Which shows it is not the option C).

Then I probed with option D. Step by step:

1) Reflection over the line y = x => (a,b) → (b,a)

2) Reflection over the x-axis => (b,a) → (b,-a)

3) Reflection over the line y = x => (b,-a) → (-a,b)

4) Reflection over the y-axis => (-a,b) → (a,b).

So, this set of reflections, given by the option D) transforms any point into itself, which proofs that the option D) is the right answer.