Triangle \( R S T \) is rotated \( 180^{\circ} \) about the origin, and then translated up 3 units. Which congruency statement describes the figures? \( \Delta R S T \cong \triangle A C B \) \( \triangle R S T \sqrt{\curvearrowleft} \triangle A B C \) \( \triangle R S T \cong \triangle B C A \)

To determine the correct congruency statement for triangle \( R S T \) after it is rotated \( 180^{\circ} \) about the origin and then translated up 3 units, we can analyze the transformations step by step. 1. **Rotation**: Rotating triangle \( R S T \) \( 180^{\circ} \) about the origin will result in a triangle \( R' S' T' \) where each vertex is transformed as follows: - If \( R(x_1, y_1) \), then \( R'(-x_1, -y_1) \) - If \( S(x_2, y_2) \), then \( S'(-x_2, -y_2) \) - If \( T(x_3, y_3) \), then \( T'(-x_3, -y_3) \) 2. **Translation**: After the rotation, translating the triangle up 3 units means adding 3 to the y-coordinates of each vertex: - \( R'(-x_1, -y_1 + 3) \) - \( S'(-x_2, -y_2 + 3) \) - \( T'(-x_3, -y_3 + 3) \) 3. **Congruency Statement**: The resulting triangle after these transformations will be congruent to the original triangle \( R S T \) because rotations and translations are rigid transformations that preserve the shape and size of the triangle. Now, let's analyze the given congruency statements: - \( \Delta R S T \cong \triangle A C B \) - \( \triangle R S T \sqrt{\curvearrowleft} \triangle A B C \) (This notation is unclear and not standard for congruency.) - \( \triangle R S T \cong \triangle B C A \) Since the transformations do not change the size or shape of the triangle, the correct congruency statement should reflect that \( \Delta R S T \) is congruent to the triangle formed after the transformations, which we can denote as \( \triangle B C A \) (assuming \( B C A \) represents the triangle after the transformations). Thus, the correct congruency statement is: \[ \Delta R S T \cong \triangle B C A \]