Describe fully the single
transformation that takes
shape A to shape B.

1. Rotate the shape 180 degrees around (9, 7.5)

2. Dilate the shape by a factor of 1/2 around the origin

Step-by-step explanation:

First, we can notice that the shape shrinks by 1/2 from shape A to sgape B -- the longest length goes from 3 to 1.5 units, and the height goes from 4 to 2. Therefore, the final step in the transformation consists of a dilation with a scale factor of 1/2 with respect to the origin to get to B.

Next, we can notice that the part sticking out of the rest of the shape goes from bottom right to top left. This represents a 180 degree rotation. If we were to rotate around the origin, however, the shape would go to the third quadrant when we want it to stay in the first.

Now, we must go back to our dilation. Before the dilation, our points will be the points on B but multiplied by 2 (B').

Here are a few points on B:

Top left, or the corner on the edge sticking out: (7,7)

Top right, or the corner horizontally opposite the edge sticking out: (8.5, 7)

Bottom left, or the corner vertically opposite the edge sticking out: (7.5, 5)

Bottom right, or the corner opposite the edge sticking out: (8.5, 5)

The points on B multiplied by 2 (B'):

Top left, or the corner on the edge sticking out: (14,14)

Top right, or the corner horizontally opposite the edge sticking out: (17, 14)

Bottom left, or the corner vertically opposite the edge sticking out: (15, 10)

Bottom right, or the corner opposite the edge sticking out: (17, 10)

Here are the same points on A:

The corner on the edge sticking out: (4, 1)

The corner horizontally opposite the edge sticking out: (1,1)

The corner vertically opposite the edge sticking out: (3, 5)

The corner opposite the edge sticking out: (1,5)

We know the last step, and we know where to start. To get from A to B', we must rotate the figure 180 degrees around a point, (x, y). To do this, we must perform the following transformation for a point (a, b) on shape A:

(a, b) -> (a-x, b-y) -> (x-a, y-b) -> (x-a+x, y-b + y) = (2x-a, 2y-b).

When rotating around a point that is not the origin, we first subtract the point we're rotating around, (x, y). Then, we apply the rotation as if we're rotating around the origin. Finally, we add (x, y) back in. For a 180 degree rotation, we turn (a-x, b-y) into (-(a-x), -(b-y)) = (x-a, y-b)

Therefore, for our points from A to B':

From (4, 1) to (14, 14):

2x - 4 = 14

add 4 to both sides to isolate the variable and its coefficient

2x = 18

divide both sides by 2 to solve for x

x = 9

2y - 1 = 14

add 1 to both sides to isolate 2y

2y = 15

divide both sides by 2 to solve for y

y = 7.5

Checking with the other points to confirm this:

From (1,1) to (17, 14):

2x - 1 = 17

x = 9

2y - 1 = 14

y = 7.5

From (3, 5) to (15, 10):

2x - 3 = 15

x = 9

2y - 5 = 10

y = 7.5

From (1, 5) to (17, 10):

2x - 1 = 17

x = 9

2y - 5 = 10

y = 7.5

Therefore, it is confirmed that we must rotate shape A around the point (9, 7.5) 180 degrees to get to B', and to get to B' to B, we must dilate the shape by a factor of 1/2 around the origin

Our transformation can be written as follows:

1. Rotate the shape 180 degrees around (9, 7.5)

2. Dilate the shape by a factor of 1/2 around the origin