Triangle xyz with vertices x(0, 0), y(0, –2), and z(–2, –2) is rotated to create the image triangle x'(0, 0), y'(2, 0), and z'(2, –2). which rules could describe the rotation? check all that apply. r0, 90° r0, 180° r0, 270° (x, y) → (–y, x) (x, y) → (y, –x)

The correct options are

(A) Rotation 90° anticlockwise.

(D)  (x, y) → (–y, x).

Step-by-step explanation:  Given that ΔXYZ is rotated to create the image triangle ΔX'Y'Z'.

Triangle XYZ and its image triangle X'Y'Z' are shown in the attached figure.

The co-ordinates of the vertices of ΔXYZ are X(0, 0), Y(0, -2) and Z(-2, -2).

And the co-ordinates of the vertices ΔX'Y'Z' are X'(0, 0), Y'(2, 0) and Z'(2, -2).

Option (A) Rotation 90°:

We see from the figure that if we rotate ΔXYZ is rotated 90° anticlockwise, then it will coincide with ΔX'Y'Z'.

So, rotation of 90° anticlockwise is a correct option.

Option (B) Rotation 180°:

If we rotate ΔXYZ is rotated clockwise or anticlockwise 180°, then it will NOT coincide with ΔX'Y'Z'.

So, rotation of 180° is NOT a correct option.

Option (C) Rotation 270°:

If we rotate ΔXYZ is rotated clockwise 270°, then also it will not coincide with ΔX'Y'Z'.

So, rotation of 270° clockwise is also a correct option.

Option (D) (x, y) → (–y, x):

We see that the co-ordinates of both the triangle follow the transformation

X(0, 0)   ⇒  X'(0, 0)

Y(0, -2)  ⇒   Y'(2, 0)

Z(-2, -2)  ⇒   Z'(2, -2).

So, the transformation is (x, y) ⇒  (-y, x).

Therefore, the  transformation (x, y) → (–y, x) is a correct option.

Option (E) (x, y) → (y, -x):

We see that the co-ordinates of both the triangle does NOT follow this transformation

For example, suppose this transformation is correct. Then, we have

Y(0, -2)  ⇒  (-2, 0), which are not the co-ordinates of Y'.

Therefore, the  transformation (x, y) → (–y, x) is NOT a correct option.

Thus, the correct options are:

(A) Rotation 90° anticlockwise.

(D)  (x, y) → (–y, x).