Respuesta :
Answer:
The single transformation that maps triangle r to triangle p is +90° (countrer clockwise) rotation about the point (-1, 0)
Explanation:
The coordinates change from reflection across y = -x is given as follows;
p = (x, y) becomes q = (-y, -x)
The reflection across the line x = -1 = reflection across the y-axis at x = -1
Reflection across the y-axis gives;
(x, y) → (-x, y)
The reflection across the line x = -1 gives;
(x, y) → (-x - 1, y)
Therefore;
q = (-y, -x) becomes r = (y -1, -x)
Hence the total transformation is presented as follows;
p = (x, y) becomes r = (y -1, -x)
y = -1 - x
Rotation of a coordinate about a point
x' = x×cosθ - y×sinθ
y' = y×cosθ + x×sinθ
Whereby we rotate -90° about the point (-1, 0), we have;
x = x'×cos(-90) - y'×sin(-90) -1 = y' - 1
y = y'×cos(-90) + x'×sin(-90) + 0 = -x'
Therefore, the single transformation that maps triangle p to triangle r is -90° rotation about the point (-1, 0)
Conversely, the single transformation that maps triangle r to triangle p is 90° rotation about the point (-1, 0).
[tex]x = x'×cos(-90) - y'×sin(-90) -1 = y' - 1y = y'×cos(-90) + x'×sin(-90) + 0 = -x'[/tex]The single transformation that maps triangle r to triangle p is about the point (-1, 0).
"Reflection"
Coordinates change from reflection y = -x .
p = (x, y) becomes q = (-y, -x)
reflection across the line x = -1
reflection across the y-axis at x = -1
Reflection across the y-axis =(x, y) → (-x, y)
The reflection across the line x = -1 gives;
(x, y) → (-x - 1, y)
q = (-y, -x) becomes r = (y -1, -x)
Now, p = (x, y) becomes r = (y -1, -x)
y = -1 - x
Rotation of a coordinate about a point :
[tex]x' = x×cosθ - y×sinθ[/tex]
[tex]y' = y×cosθ + x×sinθ[/tex]
Rotate -90° about the point (-1, 0),
[tex]x = x'×cos(-90) - y'×sin(-90) -1 = y' - 1[/tex]
[tex]y = y'×cos(-90) + x'×sin(-90) + 0 = -x'[/tex]
The single transformation that maps triangle r to triangle p is 90° rotation about the point (-1, 0).
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