To determine which of the given transformations are isometries, we need to understand what constitutes an isometry. An isometry is a transformation in geometry that preserves distances between points. Common types of isometries include translations, rotations, reflections, and glide reflections. Scaling transformations, however, do not preserve distances and thus are not isometries.
Let's analyze each of the given transformations step-by-step:
### Option (a) [tex]\( G(x, y) \rightarrow (x-2, y-2) \)[/tex]
This transformation translates the point [tex]\((x, y)\)[/tex] by moving it 2 units to the left and 2 units down. Translation does not change the distance between any points, hence it is an isometry.
### Option (b) [tex]\( H(x, y) \rightarrow (-y, x) \)[/tex]
This transformation rotates the point [tex]\((x, y)\)[/tex] 90 degrees counterclockwise around the origin. Rotation preserves the distance between points, thus this transformation is also an isometry.
### Option (c) [tex]\( H(x, y) \rightarrow (x, 4y) \)[/tex]
This transformation scales the [tex]\(y\)[/tex]-coordinate by a factor of 4 while leaving the [tex]\(x\)[/tex]-coordinate unchanged. Scaling changes the distances between points unless the scale factor is 1, so this is not an isometry.
### Option (d) [tex]\( D(x, y) \rightarrow (3x, 3y) \)[/tex]
This transformation scales both coordinates by a factor of 3. As scaling changes the distance between points, this transformation is not an isometry.
### Option (e) [tex]\( D(x, y) \rightarrow (x+\frac{1}{2}, y+\frac{1}{2}) \)[/tex]
This transformation translates the point [tex]\((x, y)\)[/tex] by moving it [tex]\(\frac{1}{2}\)[/tex] unit to the right and [tex]\(\frac{1}{2}\)[/tex] unit up. Translation preserves distances, so it is an isometry.
Based on the analysis:
- The transformations in options (a), (b), and (e) are isometries.
- The transformations in options (c) and (d) are not isometries because they involve scaling which changes distances.
Thus, the transformations that are isometries are:
- (a) [tex]\( G(x, y) \rightarrow (x-2, y-2) \)[/tex]
- (b) [tex]\( H(x, y) \rightarrow (-y, x) \)[/tex]
- (e) [tex]\( D(x, y) \rightarrow (x+\frac{1}{2}, y+\frac{1}{2}) \)[/tex]
