Question

Which of the following compositions of transformations will always produce the same image, regardless of the order in which the transformations are performed?

A.translation and reflection

B.reflection and rotation C.translation and rotation D.none of the above

Answer 1

Starting with the basics: a translation means moving the shape to a different location on the graph. Reflection means taking the figure mirroring it across the line given. A rotation means rotating the shape a certain angle so Probably C or D since if the question is asking for a similar or congruent shape

Answer 2

D. None of the above

Explanation:

Let us assume the transformations applied as follows:

Translation of h units to right and k units up i.e. (x,y) → (x+h,y+k).

Rotation of the figure around 90° i.e. (x,y) → (y,-x).

Reflection of the figure about x-axis i.e. (x,y) → (x,-y).

Now, we need to find the composition of transformations that gives the same image irrespective of the order.

So, let us apply the above assumed transformations on the point (x,y). Therefore, according to options:

A. Translation then reflection gives (x,y) → (x+h, y+k) → (x+h,-y-k)

or Reflection then translation will give (x,y) → (x,-y) → (x+h, -y-k).

B. Reflection then Rotation gives (x,y) → (x,-y) → (-y,-x)

or Rotation then reflection gives (x,y) → (y,-x) → (y,x)

C. Translation then rotation gives (x,y) → (x+h,y+k) → (y+k,-x-h)

Rotation then translation will give (x,y) → (y,-x) → (y+k,-(x+h) = (y+k,-x-h).

As, we can see that the resulting figure depends on the order in which the rigid transformation is applied and does not gives us the original image.

So, option D is correct.