Which of the following statements is true about transformations in geometry?

Which of the following statements is true about transformations in geometry?

• A. They can change the size of the figure.
• B. They always keep the shape and size of the figure unchanged.
• C. They can involve translations only.
• D. They are limited to two-dimensional figures.

Answer: (B)

The statement that transformations always keep the shape and size of the figure unchanged is true for a specific category of transformations known as rigid transformations or isometries. These transformations include translations, rotations, and reflections. Each of these transformations alters the position of a figure in the plane without changing its size or shape. For instance, a triangle can be moved (translated) or flipped (reflected) while maintaining the same angle measures and side lengths. The other statements do not accurately reflect the properties of rigid transformations. While it is correct that transformations can change size, as seen in dilation (a type of transformation that makes a figure larger or smaller), this does not apply to all transformations. Similarly, stating that transformations can involve translations only is incorrect because transformations can include various types such as rotations and reflections. Lastly, transformations are not limited to two-dimensional figures; they also apply to three-dimensional figures, which can undergo transformations like rotations and translations in three-dimensional space. Thus, the accurate understanding of rigid transformations supports the truth of the selected statement.

The statement that transformations always keep the shape and size of the figure unchanged is true for a specific category of transformations known as rigid transformations or isometries. These transformations include translations, rotations, and reflections. Each of these transformations alters the position of a figure in the plane without changing its size or shape. For instance, a triangle can be moved (translated) or flipped (reflected) while maintaining the same angle measures and side lengths.

The other statements do not accurately reflect the properties of rigid transformations. While it is correct that transformations can change size, as seen in dilation (a type of transformation that makes a figure larger or smaller), this does not apply to all transformations. Similarly, stating that transformations can involve translations only is incorrect because transformations can include various types such as rotations and reflections. Lastly, transformations are not limited to two-dimensional figures; they also apply to three-dimensional figures, which can undergo transformations like rotations and translations in three-dimensional space. Thus, the accurate understanding of rigid transformations supports the truth of the selected statement.