2. Select all of the following transforr nations that preserve distance. A Reflectio mover the y-axis B. (x,y)arrow (6x,6y) C.Rotation 90^circ clockwise about the origin D (x,y)arrow (x,-y) E. Dilat sc ale factor 0.5 centered at the o rigin F (x,y)arrow (x-8,y+3)

## Step 1<br />### The problem involves understanding the concept of transformations in mathematics. A transformation is a function that changes the position, shape, and/or size of a figure. There are four main types of transformations: translations (slides), reflections (flips), rotations (turns), and dilations (identical shape, but different size). A transformation that preserves distance is also known as isometry. <br /><br />## Step 2<br />### We need to evaluate each transformation to determine whether it preserves distance. <br /><br />## Step 3<br />### A) Reflection over the y-axis is an example of isometry as it preserves the distance between each point and its image.<br /><br />## Step 4<br />### B) The function (x, y) → (6x, 6y) is a dilation transformation and not isometric, where all distances are multiplied by the same factor, but the original distances among the points are not preserved.<br /><br />## Step 5<br />### C) Rotation 90° clockwise about the origin is another example of isometry: the distance from the origin to a point and its image remains constant, as does the distance between any pair of points and their images.<br /><br />## Step 6<br />### D) The transformation (x,y) to (x,-y) reflects a figure through the x-axis, which again is isometric since it simply flips a figure while preserving the original distances.<br /><br />## Step 7<br />### E) Dilation with scale factor 0.5 centered at the origin, like any dilation that isn't with factor 1, doesn't preserve distances among original points and is hence not isometric.<br /><br />## Step 8<br />### F) In the transformation (x, y) → (x-8, y+3), known as translation since it slides or moves every point the same distance in the stated direction, regardless of their original positions, is another instance of isometry.<br /><br />## Step 9<br />### Based on these interpretations of each transformation, isometric transformations from the given options are reflection over the y-axis, rotation 90° clockwise about the origin, reflection over x-axis (from D option) and translation (from F).