Question

Triangle XYZ is rotated 90° counterclockwise using the origin as the center of rotation.

On a coordinate plane, triangle X Y Z has points (2, 2), (4, 2), (1, negative 1). Triangle X prime Y prime Z prime has points (negative 2, 2), (negative 2, 4), (1, 1).

Which other rotation can be used to create triangle X’Y’Z’ from triangle XYZ?
90° clockwise
270° clockwise
270° counterclockwise
360° counterclockwise

Answer 1

To determine which rotation can transform triangle XYZ to triangle X'Y'Z', we can analyze the transformations given.

Triangle XYZ has the following points:
- \( X(2, 2) \)
- \( Y(4, 2) \)
- \( Z(1, -1) \)

Triangle X'Y'Z' has these points:
- \( X'(-2, 2) \)
- \( Y'(-2, 4) \)
- \( Z'(1, 1) \)

Now, let's find the coordinates of each point in triangle XYZ after a 90° clockwise rotation.

### Rotating Triangle XYZ
To rotate a point \( (x, y) \) 90° clockwise around the origin, the coordinates change according to the rule:
\[
(x, y) \rightarrow (y, -x)
\]

Applying this rule to each point:

1. For point \( X(2, 2) \):
\[
X \rightarrow (2, -2)
\]

2. For point \( Y(4, 2) \):
\[
Y \rightarrow (2, -4)
\]

3. For point \( Z(1, -1) \):
\[
Z \rightarrow (-1, -1)
\]

This does not match any of the points in triangle X'Y'Z'.

### Checking Other Rotations
Next, let's check the effect of a 270° clockwise rotation. The 270° clockwise rotation is equivalent to a 90° counterclockwise rotation. The coordinates change according to the rule:
\[
(x, y) \rightarrow (-y, x)
\]

Applying this rule to each point:

1. For point \( X(2, 2) \):
\[
X \rightarrow (-2, 2)
\]

2. For point \( Y(4, 2) \):
\[
Y \rightarrow (-2, 4)
\]

3. For point \( Z(1, -1) \):
\[
Z \rightarrow (1, 1)
\]

Thus, after a 270° clockwise rotation, we obtain the coordinates for triangle X'Y'Z':
- \( X'(-2, 2) \)
- \( Y'(-2, 4) \)
- \( Z'(1, 1) \)

This matches triangle X'Y'Z'.

### Conclusion
The other rotation that can be used to create triangle X'Y'Z' from triangle XYZ is **270° clockwise**.