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Rotate the figure [tex]$270^{\circ}$[/tex] clockwise about the origin. Then decide if each statement about rotated figures is true or false.

\begin{tabular}{|c|c|c|}
\hline
Statement about rotated figures & True & False \\
\hline
\begin{tabular}{l}
If two sides are parallel in the original figure, then those sides may not be parallel in the final figure.
\end{tabular} & & \\
\hline
\begin{tabular}{l}
The final side lengths are the same as the original side lengths.
\end{tabular} & & \\
\hline
\begin{tabular}{l}
The final angle measures are smaller than the original angle measures.
\end{tabular} & & \\
\hline
\begin{tabular}{l}
The original figure and the final figure may not be congruent.
\end{tabular} & & \\
\hline
\end{tabular}



Answer :

GinnyAnswer
To answer the question about the properties of the rotated figure, we'll analyze each statement and determine whether it is true or false based on the principles of rotation in geometry.

To begin, let's understand what a 270° clockwise rotation about the origin does. This specific rotation will map any point [tex]\((x, y)\)[/tex] on the original figure to the point [tex]\((y, -x)\)[/tex] on the rotated figure. It’s important to note that rotations are rigid transformations, meaning they preserve distances, angles, and parallelism.

Now, let's examine each statement.

1. Statement: If two sides are parallel in the original figure, then those sides may not be parallel in the final figure.
- Analysis: Rotations preserve the orientation and parallelism of lines. If two sides are parallel before the rotation, they will remain parallel after the rotation.
- Conclusion: This statement is false.\
Answer: False

2. Statement: The final side lengths are the same as the original side lengths.
- Analysis: A rotation is a distance-preserving transformation (also known as an isometry). Therefore, the lengths of sides remain unchanged.
- Conclusion: This statement is true.\
Answer: True

3. Statement: The final angle measures are smaller than the original angle measures.
- Analysis: Rotations preserve angle measures. The angles in the original figure and the rotated figure will be exactly the same.
- Conclusion: This statement is false.\
Answer: False

4. Statement: The original figure and the final figure may not be congruent.
- Analysis: As previously mentioned, rotations are rigid transformations. This means that they produce congruent figures, preserving size and shape.
- Conclusion: This statement is false.\
Answer: False

Putting it all together, we have the following answers for each statement:
- If two sides are parallel in the original figure, then those sides may not be parallel in the final figure: False
- The final side lengths are the same as the original side lengths: True
- The final angle measures are smaller than the original angle measures: False
- The original figure and the final figure may not be congruent: False

Thus, the final result is:

[tex]\[(\text{False}, \text{True}, \text{False}, \text{False})\][/tex]

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