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Transformations: Reflections

Triangle [tex]A B C[/tex] has coordinates [tex]A(1,-1), B(0,2)[/tex], and [tex]C(2,1)[/tex] and it is reflected over the line [tex]y=-x[/tex] to form triangle [tex]A^{\prime} B^{\prime} C^{\prime}[/tex]. What are the coordinates of triangle [tex]A^{\prime} B^{\prime} C^{\prime}[/tex]?

Sagot :

GinnyAnsweridn
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]
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