Compositions of Transformations Quick Check

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Question
Use the image to answer the question.

Two quadrilaterals, upper A upper B upper C upper D and upper W upper X upper Y upper Z, are graphed on a coordinate plane. The x-axis ranges from negative 14 to 10 in increments of 2. The y-axis ranges from negative 6 to 10 in increments of 2.

What two rigid transformations were performed on figure ABCD to produce the congruent figure WXYZ?

(1 point)
Responses choose the correct answer carefully and tell me why this is the true correct answer

ABCD was first rotated −270° (clockwise), then shifted 3 units to the left and 5 units up.
upper A upper B upper C upper D was first rotated negative 270 degrees (clockwise), then shifted 3 units to the left and 5 units up.

ABCD was first rotated 90° (counterclockwise), then shifted 3 units to the left and 5 units up.
upper A upper B upper C upper D was first rotated 90 degrees (counterclockwise), then shifted 3 units to the left and 5 units up.

ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise).
upper A upper B upper C upper D was first shifted 3 units to the left and 5 units up, then rotated 90 degrees (counterclockwise).

ABCD was first shifted 3 units to the left and 5 units up, then rotated −90° (clockwise).
upper A upper B upper C upper D was first shifted 3 units to the left and 5 units up, then rotated negative 90 degrees (clockwise).
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Answer 1

To determine the correct answer about the rigid transformations performed on figure ABCD to produce the congruent figure WXYZ, we can consider the types of transformations involved: rotation and translation (shifting).

1. Analyzing the transformations:

   • Rotation: This involves turning the shape around a specific point. The degrees of rotation can be clockwise or counterclockwise.
   • Translation: This involves moving the shape in a straight direction without changing its orientation.

2. Shifts: From the problem, it states that the figure was shifted "3 units to the left and 5 units up". This movement should be a translation transformation that should appear first as we consider transformations in sequential order.

3. Rotation: The rotation options are either negative 90° (clockwise) or 90° (counterclockwise).

   • To determine the correct rotation, we can analyze if the resulting figure WXYZ appears rotated correctly in relation to figure ABCD after the translation.

Given these points, we can evaluate the response options provided in the question:

• The first option states that ABCD was first rotated −270° (clockwise), then shifted. However, a −270° (clockwise) rotation is equivalent to a 90° (counterclockwise) rotation, which means it is actually incorrect in naming the transformation correctly.

• The second option states that ABCD was first rotated 90° (counterclockwise), then shifted. This transformation order is plausible, but we would need confirming visual details to ensure the translation matches after rotating first.

• The third option states that ABCD was first shifted, then rotated 90° (counterclockwise). This would suggest that the translation happened first, followed by a corresponding rotation, which is often the common sequence when movements are involved.

• The last option proposes that ABCD was first shifted and then rotated −90° (clockwise); it’s important to realize that this is effectively a rotation that could result in similar positioning as well.

4. Final Choice: Considering all of these elements, the best and most logical answer would be the third option, which indicates that "ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise)". This order complies with usual transformation rules, and translates logically into getting figure WXYZ from ABCD by maintaining congruency after both transformations.

Thus, the correct response is: "ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise)."