If (a,b) is reflected in the x-axis, then it's image is the point (a,- b).
If (a,b) is reflected in the y-axis, then it's image is the point (- a,b).
If (a,b) is reflected in the line y=x, then it's image is the point (b,a).
If (a,b) is reflected in the line y=- x, then it's image is the point (- b,- a).
Yes, it is possible. Explanation: See solution.
Practice makes perfect
As long as the two reflections result in the same transformations as a rotation, we know that the rotations can be written as the composition of two reflections.
90^(∘) Rotation
When we rotate a figure 90^(∘) counterclockwise, the vertices will change in the following way.
( a, b) → (- b, a)
Using a combination of reflections, we can accomplish this transformation in two ways:
First we interchange the x- and y-coordinates. Second, we change signs of the x-coordinate.( a, b) → ( b, a) → (- b, a)
First, we change signs of the y-coordinate. Second, we interchange the x- and y-coordinates.( a, b) → ( a,- b) → (- b, a)
Both sets of transformation will mimic the same behavior as if we rotated the figure by 90^(∘). If we look at the first set of transformations, we can interchange the x- and y-coordinate if we reflect the figure in y=x. Second, to change the sign of the x-coordinate we reflect it in the y-axis.
The second set of transformations require us to first change the sign of the y-coordinate. We can do that by reflecting the figure in the x-axis. Finally we have to interchange the x- and y-coordinates which we know we can accomplish by reflecting the figure in y=x.
180^(∘) Rotation
When we rotate a figure 180^(∘) counterclockwise, the vertices will change in the following way.
( a, b) → (- a,- b).
Using a combination of reflections, we can accomplish this transformation in two ways:
First we change sign of the x-coordinate and then we change sign of the y-coordinate (or vice verse).( a, b) → (- a, b) → (- a,- b)
First we interchange places of the x-and y-coordinate and change the sign of both coordinates. Second, we interchange the x- and y-coordinates once again. ( a, b) → (- b,- a) → (- a,- b).
Both sets of transformation will mimic the same behavior as if we rotated the figure by 180^(∘). If we look at the first set of transformation, we can change sign of the x-coordinate by reflecting the figure in the y-axis. Second, we can change the sign of the y-coordinate by reflecting the figure in the x-axis.
The second set of transformations require us to both interchange the x- and y-coordinates and change their signs. This can be accomplished by a reflection in y=- x. Finally, we have to interchange the x- and y-coordinates again which we can accomplish by reflecting it in y=x.
270^(∘) Rotation
When we rotate a figure 270^(∘) counterclockwise, the vertices will change in the following way.
( a, b) → ( b,- a).
Using a combination of reflections, we can accomplish this transformation in two ways:
First we change the sign of the x-coordinate. Second, we interchange the x- and y-coordinate.( a, b) → (- a, b) → ( b,- a)
First we interchange the x-and y-coordinate. Second, we change the sign of the y-coordinate.( a, b) → ( b, a) → ( b,- a).
Both sets of transformations will mimic the same behavior as if we rotated the figure by 270^(∘). If we look at the first set of transformation, we can change the sign of the x-coordinate by reflecting the figure in the y-axis. Second, we can interchange the x- and y-coordinate by reflecting the figure in y=x.
The second set of transformations require us to first interchange the x- and y-coordinates. This can be accomplished by a reflection in y=x. Finally, we have to change the sign of the y-coordinate which we know we can accomplish by reflecting it in the x-axis.
360^(∘) Rotation
A 360^(∘) rotation brings the figure back to it's original position. Using reflections, we can accomplish this by reflecting it twice in the same line of reflection.
