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△EFG is rotated −90 degrees (clockwise) to create △XYZ . Which of the following statements must be true?

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To determine which statements must be true when triangle \( \triangle EFG \) is rotated \(-90\) degrees (clockwise) to create triangle \( \triangle XYZ \), we need to understand the properties of rotation.

  1. Orientation Change: The vertices of triangle \( \triangle EFG \) will change their orientation after the rotation. Rotating \(-90\) degrees clockwise will change the order of the points, typically altering their appearance while still maintaining the shape and size of the triangle.

  2. Vertex Correspondence: If we denote the points before the rotation as \( E, F, G \), and after the rotation as \( X, Y, Z \), we need to determine how the points correspond. After a \(-90\) degrees rotation around a point (usually the origin), each point \( (x, y) \) will map to \( (y, -x) \).

  3. Angle Measures: The interior angles of triangle \( \triangle EFG \) will remain the same in triangle \( \triangle XYZ \) after rotation since rotations do not change angle measures.

  4. Congruence: The triangles \( \triangle EFG \) and \( \triangle XYZ \) will be congruent since rotation is an isometric transformation that preserves distance and shape.

  5. Side Lengths: The corresponding sides of \( \triangle EFG \) and \( \triangle XYZ \) will be of equal length.

With this in mind, consider the following statements to determine which must be true:

  • Statement 1: The sides of \( \triangle XYZ \) are equal to the corresponding sides of \( \triangle EFG \).

    • This is true because rotation preserves lengths.

  • Statement 2: The angles of \( \triangle XYZ \) are equal to the corresponding angles of \( \triangle EFG \).

    • This is also true because rotation preserves angle measures.

  • Statement 3: The order of the vertices of \( \triangle EFG \) is the same as that of triangle \( \triangle XYZ \).

    • This is false. The order of vertices changes due to the rotation.

  • Statement 4: \( \triangle XYZ \) has the same orientation as \( \triangle EFG \).

    • This is false since rotation changes the orientation.

Conclusion

The statements that must be true after rotating triangle \( \triangle EFG \) \(-90\) degrees clockwise to form triangle \( \triangle XYZ \) are about congruence of sides and angles, while the corresponding vertex order and orientation will be different.