Question

What set of reflections would carry triangle ABC onto itself?

Answer 1

The correct answer is:
4) y-axis, x-axis, y-axis, x-axis.

Explanation:
Reflecting a point (x,y) across the x-axis will map it to (x,-y).
Reflecting a point (x,y) across the y-axis will map it to (-x,y).
Reflecting a point (x,y) across the line y=x will map it to (y, x).

We want a series of transformations that will map every point (x,y) back to (x,y). This means that everything that gets done in one transformation must be undone in another. The only one where this happens is #4.

Reflecting across the y-axis first negates the x-coordinate; (x,y) goes to (-x,y).
Reflecting this across the x-axis negates the y-coordinate; (-x,y) goes to (-x,-y).
Reflecting this point back across the y-axis negates the x-coordinate again, returning it to the original: (-x,-y) goes to (x,-y).
Reflecting this point back across the x-axis negates the y-coordinate again, returning it to the original: (x,-y) goes to (x,y).
We are back to our original point.

Answer 2

Answer: 4) y-axis, x-axis, y-axis, x-axis

Step-by-step explanation:

Let (x,y) represents the coordinates of the triangle ABC,

1) Under the set of reflection, x-axis, y=x, y-axis, x-axis,

$(x,y)\rightarrow (x,-y)\rightarrow (-x,-(-y))\rightarrow (-(-x),y)\rightarrow (x,-y)$

Hence, (x,-y) represents the coordinates of resultant transformed triangle.

⇒ This set of reflections does not carry triangle ABC onto itself.

2) Under the set of reflection,  x-axis, y-axis, x-axis

$(x,y)\rightarrow (x,-y)\rightarrow (-x,-y))\rightarrow (-x,y)$

Hence, (-x,y) represents the coordinates of resultant transformed triangle.

⇒ This set of reflections does not carry triangle ABC onto itself.

3) Under the set of reflection, y=x, x-axis, x-axis

$(x,y)\rightarrow (-x,-y)\rightarrow (-x,-(-y))\rightarrow (-x,-y)$

Hence, (-x,-y) represents the coordinates of resultant transformed triangle.

⇒ This set of reflections does not carry triangle ABC onto itself.

4) Under the set of reflection, y-axis, x-axis, y-axis, x-axis

$(x,y)\rightarrow (-x,y)\rightarrow (-x,-y)\rightarrow (-(-x),-y)\rightarrow (x,-(-y))$

Hence, (x,y) represents the coordinates of resultant transformed triangle.

⇒ This set of reflections carries triangle ABC onto itself.