To analyze the information provided about the rotation of triangle EFG to create triangle XYZ, let's break down the effect of a -90 degrees (clockwise) rotation on the vertices.
When a point \((x, y)\) is rotated -90 degrees clockwise around the origin, its coordinates change according to the following transformation:
\[ (x, y) \rightarrow (y, -x) \]
Assuming E, F, and G are at positions in the plane, we will denote their original coordinates as follows:
- \( E (x_1, y_1) \)
- \( F (x_2, y_2) \)
- \( G (x_3, y_3) \)
After applying the -90 degrees rotation:
- \( E' (y_1, -x_1) \) → corresponds to \( X \)
- \( F' (y_2, -x_2) \) → corresponds to \( Y \)
- \( G' (y_3, -x_3) \) → corresponds to \( Z \)
Looking at the potential truths about the relationships between the segments and points in the rotated shape XYZ, we can analyze each option:
a. \( EF \) and \( XZ \): This cannot be determined as a true statement without additional context on the lengths, so it does not have to be true.
b. \( EG \) and \( XY \): Similar reasoning as above can be applied. There is no inherent relationship that must hold true.
c. \( E \) and \( X \): This relationship must be true since point E is rotated to point X during the transformation.
d. \( F \) and \( Z \): This is not necessarily true based on the rotation described.
Given the above reasoning, the only statement that must hold true is:
c. E X
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