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 reflectionb.reflection and rotationc.translation and rotationd.none of the above

D. None of the above              

Step-by-step 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.

Complete question is;

Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measures to the nearest degree.

a = 9, b = 7, A = 108°

c = 3.8

C = 24°

B = 48°

Step-by-step explanation:

From law of sines, we can find B.

a/sin A = b/sin B

sin B = (bsinA)/a

sin B = (7 × sin 108)/9

sin B = 0.7397

B = sin^(-1) 0.7397

B ≈ 48°

Sum of angles in a triangle = 180°

Thus;

C = 180 - (A + B)

C = 180 - (108 + 48)

C = 24°

Again, using law of sines, we have;

a/sin A = c/sin C

c = (a•sin C)/(sin A)

c = (9 × sin 24)/(sin 108)

c = 3.849

To the nearest tenth gives;

c = 3.8