Final answer:
In geometry, transformations that preserve a figure's lengths and angle measures include horizontal translation, reflection over the x-axis, and rotation of 90° about the origin. Horizontal and vertical stretches are not isometries and hence do not preserve these properties.
Step-by-step explanation:
In mathematics, particularly in geometry, various transformations can be applied to figures. These transformations can either preserve or alter certain properties of the figures, such as lengths and angle measures. When considering the transformations that preserve a figure’s lengths and angle measures, it is important to distinguish between those that are isometries and those that are not.
The following transformations are isometries and thus preserve a figure’s lengths and angle measures:
- Horizontal translation: This involves moving a figure horizontally without changing its size or orientation.
- Reflection over the x-axis: Reflecting a figure over the x-axis results in a mirror image, but does not alter lengths or angles.
- Rotation of 90° about the origin: Rotating a figure 90 degrees around the origin changes the figure’s position but its shape, size, and angle measures remain the same.
The following are not isometries and do not preserve the lengths and angles of a figure:
- Horizontal stretch: This transformation alters the figure by stretching it horizontally, changing the lengths but not the angle measures.
- Vertical stretch: Similar to a horizontal stretch, this changes the vertical lengths of a figure.
Therefore, the correct options that preserve a figure's lengths and angle measures are horizontal translation, reflection over the x-axis, and rotation of 90° about the origin.