To determine which rotation has been applied to triangle [tex]\( RST \)[/tex] to obtain the image triangle [tex]\( R'S'T' \)[/tex], we need to check how the coordinates of the vertices change under each possible rotation.
Here are the vertices for the original and the rotated triangles:
- Original vertices are:
- [tex]\( R (2, 0) \)[/tex]
- [tex]\( S (4, 0) \)[/tex]
- [tex]\( T (1, -3) \)[/tex]
- Rotated vertices are:
- [tex]\( R' (0, -2) \)[/tex]
- [tex]\( S' (0, -4) \)[/tex]
- [tex]\( T' (-3, -1) \)[/tex]
We need to determine which rotation maps these original points to their corresponding rotated points. Let us consider the following possibilities:
1. Rotation [tex]\( 90^\circ \)[/tex] Counterclockwise:
- The rule for [tex]\( 90^\circ \)[/tex] rotation is [tex]\( (x, y) \rightarrow (-y, x) \)[/tex].
- Applying this:
- [tex]\( R (2, 0) \rightarrow (0, 2) \)[/tex]
- [tex]\( S (4, 0) \rightarrow (0, 4) \)[/tex]
- [tex]\( T (1, -3) \rightarrow (3, 1) \)[/tex]
- These do not match [tex]\( R' (0, -2) \)[/tex], [tex]\( S' (0, -4) \)[/tex], [tex]\( T' (-3, -1) \)[/tex].
2. Rotation [tex]\( 180^\circ \)[/tex] Counterclockwise:
- The rule for [tex]\( 180^\circ \)[/tex] rotation is [tex]\( (x, y) \rightarrow (-x, -y) \)[/tex].
- Applying this:
- [tex]\( R (2, 0) \rightarrow (-2, 0) \)[/tex]
- [tex]\( S (4, 0) \rightarrow (-4, 0) \)[/tex]
- [tex]\( T (1, -3) \rightarrow (-1, 3) \)[/tex]
- These do not match [tex]\( R' (0, -2) \)[/tex], [tex]\( S' (0, -4) \)[/tex], [tex]\( T' (-3, -1) \)[/tex].
3. Rotation [tex]\( 270^\circ \)[/tex] Counterclockwise:
- The rule for [tex]\( 270^\circ \)[/tex] rotation is [tex]\( (x, y) \rightarrow (y, -x) \)[/tex].
- Applying this:
- [tex]\( R (2, 0) \rightarrow (0, -2) \)[/tex]
- [tex]\( S (4, 0) \rightarrow (0, -4) \)[/tex]
- [tex]\( T (1, -3) \rightarrow (-3, -1) \)[/tex]
- These exactly match [tex]\( R' (0, -2) \)[/tex], [tex]\( S' (0, -4) \)[/tex], [tex]\( T' (-3, -1) \)[/tex].
4. Rotation [tex]\( 360^\circ \)[/tex] Counterclockwise:
- The rule for [tex]\( 360^\circ \)[/tex] rotation is [tex]\( (x, y) \rightarrow (x, y) \)[/tex].
- Applying this:
- [tex]\( R (2, 0) \rightarrow (2, 0) \)[/tex]
- [tex]\( S (4, 0) \rightarrow (4, 0) \)[/tex]
- [tex]\( T (1, -3) \rightarrow (1, -3) \)[/tex]
- These do not match [tex]\( R' (0, -2) \)[/tex], [tex]\( S' (0, -4) \)[/tex], [tex]\( T' (-3, -1) \)[/tex].
Since the only matching transformation is a rotation of [tex]\( 270^\circ \)[/tex] counterclockwise, the rule that describes the transformation is:
[tex]\[ \boxed{R_{0,270^\circ}} \][/tex]
