To find the coordinates of the reflected triangle [tex]\( A'B'C' \)[/tex], we need to reflect each vertex of the original triangle [tex]\( ABC \)[/tex] over the line [tex]\( y = -x \)[/tex].
When reflecting a point [tex]\((x, y)\)[/tex] over the line [tex]\( y = -x \)[/tex], the coordinates of the reflected point [tex]\((x', y')\)[/tex] can be found by swapping the coordinates and negating them:
[tex]\[
(x', y') = (-y, -x)
\][/tex]
Let's apply this transformation to each vertex of the triangle one by one:
Step-by-step Solution:
1. Reflect point [tex]\( A(1, -1) \)[/tex]:
- Original coordinates: [tex]\( (1, -1) \)[/tex]
- Swap and negate: [tex]\( (-(-1), -1) = (1, -1) \)[/tex]
- So, [tex]\( A' = (1, -1) \)[/tex]
2. Reflect point [tex]\( B(0, 2) \)[/tex]:
- Original coordinates: [tex]\( (0, 2) \)[/tex]
- Swap and negate: [tex]\( (-2, -0) = (-2, 0) \)[/tex]
- So, [tex]\( B' = (-2, 0) \)[/tex]
3. Reflect point [tex]\( C(2, 1) \)[/tex]:
- Original coordinates: [tex]\( (2, 1) \)[/tex]
- Swap and negate: [tex]\( (-1, -2) = (-1, -2) \)[/tex]
- So, [tex]\( C' = (-1, -2) \)[/tex]
Therefore, the coordinates of the reflected triangle [tex]\( A'B'C' \)[/tex] are:
[tex]\[
A'(1, -1), B'(-2, 0), C'(-1, -2)
\][/tex]
