Question

A sequence of transformations that proves congruence between shape 1 and shape 2 by mapping shape 1 onto shape 2 is a reflection across the y-axis, followed by a

A. reflection across the x-axis

B. 90-degree clockwise rotation about the origin

C. 90-degree counterclockwise rotation about the origin

D. 180-degree rotation about the origin

and then a transition 1 unit down. Alternatively, a sequence of transformations that proves congruence between shape 2 and shape 1 by mapping shape 2 onto shape 1 is a reflection of shape 2 about the x-axis followed by a

A. reflection across the y-axis

B. 90-degree clockwise rotation about the origin

C. 90-degree counterclockwise rotation about the origin

D. 180-degree rotation about the origin

and then a translation 1 unit left.

Answer 1

For Shape 1 to Shape 2: B (90-degree clockwise rotation about the origin)

For Shape 2 to Shape 1: C (90-degree counterclockwise rotation about the origin)

To map Shape 1 onto Shape 2 and then Shape 2 back onto Shape 1 with the given transformations, we must ensure that the series of transformations are inverses of each other in reverse order.

Given the first sequence for mapping Shape 1 onto Shape 2:

1. Reflection across the y-axis.

2. An additional transformation (A, B, C, or D).

3. Translation 1 unit down.

We need to find the inverse operations to map Shape 2 back onto Shape 1:

1. Translation 1 unit to the left (the inverse of translating 1 unit down is translating 1 unit up, but since we are now performing the inverse operation, it becomes 1 unit to the left).

2. The inverse of the additional transformation (A, B, C, or D).

3. Reflection across the x-axis (which is already given).

Now, let's analyze the additional transformation needed to map Shape 1 onto Shape 2:

- If you reflect Shape 1 across the y-axis, it is now a mirror image across the y-axis.

- Then if you rotate this image 90 degrees clockwise about the origin, its orientation will match that of Shape 2 if Shape 2 was originally a 90-degree counterclockwise rotation of Shape 1.

- Finally, if you translate this image 1 unit down, you will have mapped Shape 1 onto Shape 2.

Now, for the inverse sequence, starting with Shape 2:

- First, translating 1 unit left to counteract the 1 unit down translation.

- Then, the inverse transformation of a 90-degree clockwise rotation about the origin is a 90-degree counterclockwise rotation about the origin.

- Lastly, a reflection across the x-axis.

Therefore, the sequences that prove congruence between Shape 1 and Shape 2 are:

For mapping Shape 1 onto Shape 2:

1. Reflection across the y-axis.

2. 90-degree clockwise rotation about the origin (Option B).

3. Translation 1 unit down.

For mapping Shape 2 onto Shape 1:

1. Translation 1 unit left.

2. 90-degree counterclockwise rotation about the origin (Option C).

3. Reflection across the x-axis.

So the answers are:

For Shape 1 to Shape 2: B (90-degree clockwise rotation about the origin)

For Shape 2 to Shape 1: C (90-degree counterclockwise rotation about the origin)