Answer:
The point O can be (x,y) or (-x,y) or (-x,-y) or ( x,-y) reason being that you haven't given point O lies in which quadrant.
Rotation through 90° counter clockwise
(x,y) = (-y,x)
(-x,y)=(-y,-x)
(-x,-y)=(y,-x)
(x,-y) =(y,x)
Then Reflection across X axis has taken place.
(-y,x) = (-y,-x)
(-y,-x)=(-y,x)
(y,-x)=(y,x)
(y,x)=(y,-x)
Then a reflection across the y-axis has taken place.
(-y,-x)=(y,-x)
(-y,x) = (y,x)
(y,x) =(-y,x)
(y,-x)=(-y,-x)
Then a translation a units to the right and b units up has taken place.
(y,-x)=(y+a,-x+b)
(y,x) =(y+a, x+b)
(-y,x) =(-y+a,x+b)
(-y,-x)=(-y+a,-x+b)
Then a rotation of 180 degrees counterclockwise about the origin has taken place.
(y+a,-x+b)=[-(y+a),-(-x+b)]
(y+a, x+b)=[- (y+a), -(x+b)]
(-y+a,x+b)=[-(-y+a),-(x+b)]
(-y+a,-x+b) =[-(-y+a),-(-x+b)]
Now Again a reflection across the Y axis has taken place.
[-(y+a),-(-x+b)]=[(y+a),-(-x+b)]
[-(y+a),-(x+b)]=[(y+a),-(-x+b)]
[-(-y+a),-(x+b)]=[(-y+a),-(x+b)]
[-(-y+a),-(-x+b)]=[(-y+a),-(-x+b)]
Totally depends on value of a and b on which quadrant these point lies.
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