Which of the following transformations will result in an image that maps onto itself? a. rotate 180 degrees counterclockwise and then rotate 90 degrees clockwiseb. reflect across a line and then reflect again across the linec. rotate 270 degrees counterclockwise and the translate 5 units leftd. reflect across the x-axis and then reflect across the y-axis

General Idea:

Reflection Rules:

The reflection of the point (x,y) across the x-axis is the point (x,-y).

The reflection of the point (x,y) across the y-axis is the point (-x,y).

The reflection of the point (x,y) across the line is the point (y, x)

Rotation Rules:

The 90 degree counterclockwise rotation about the origin of a point (x, y) is the point (-y, x)

The 180 degree counterclockwise rotation about the origin of a point (x, y) is the point (-x, -y)

The 270 degree counterclockwise rotation about the origin of a point (x, y) is the point (y, -x)

The 90 degree clockwise rotation about the origin of a point (x, y) is the point (y, -x)

The 180 degree clockwise rotation about the origin of a point (x, y) is the point (-x, -y)

The 270 degree clockwise rotation about the origin of a point (x, y) is the point (-y, x)

Applying the concept:

Say if we have a point (x, y)

After we "Rotate 180 degrees counterclockwise" the point becomes (-x, -y) and after we perform the transformation "rotate 90 degrees clockwise" the point becomes (-y, x). So transformation described in Option A WILL NOT result in an image that maps onto itself.

After we "Reflect across a line" the point becomes (y, x) and after we "then reflect again across the line" the point (y, x) becomes (x, y). So transformation described in Option B WILL result in an image that maps onto itself.

After we "rotate 270 degrees counterclockwise" the point becomes (y, -x) and after we " translate 5 units left" the point (y, -x) becomes (y - 5, -x). So transformation described in Option C WILL NOT result in an image that maps onto itself.

After we "reflect across the x-axis" the point becomes (x, -y) and after we "reflect across the y-axis" the point (x, -y) becomes (-x, -y). So transformation described in Option D WILL NOT result in an image that maps onto itself.

Conclusion:

The answer is option B. The following transformation of "reflect across a line and then reflect again across the line" will result in an image that maps onto itself.

Area of the rectangular park = 2/3 square mile
Let us assume the width of the rectangular park = x
Then
The length of the rectangular park = 2 2/3 * x mile
                                                       = 8x/3 mile
Then
Area of the rectangular park = Length * Width
2/3 = (8x/3) * x
2/3 = 8x^2/3
(2 * 3)/3 = 8x^2
2 = 8x^2
x^2 = 2/8
x^2 = 1/4
x^2 = (1/2)^2
x = 1/2
So the width of the park is 1/2 feet.