(7) Shanika draws figure \( A B C D \) in the coordinate plane. She rotates figure \( A B C D 180^{\circ} \) clockwise around the origin to form figure \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \). a. Draw figure \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \). Compare corresponding \( x \)-coordinates of the vertices of figures \( A B C D \) and \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \). Then, compare corresponding \( y \)-coordinates.

To rotate a figure \( ABCD \) 180 degrees clockwise around the origin, we can use the following transformation for the coordinates of each vertex: - If a point has coordinates \( (x, y) \), after a 180-degree rotation, its new coordinates will be \( (-x, -y) \). Let's denote the coordinates of the vertices of figure \( ABCD \) as follows: - \( A(x_1, y_1) \) - \( B(x_2, y_2) \) - \( C(x_3, y_3) \) - \( D(x_4, y_4) \) After rotating 180 degrees clockwise, the coordinates of the vertices of figure \( A'B'C'D' \) will be: - \( A'(-x_1, -y_1) \) - \( B'(-x_2, -y_2) \) - \( C'(-x_3, -y_3) \) - \( D'(-x_4, -y_4) \) ### Comparison of Coordinates 1. **Comparison of \( x \)-coordinates:** - For each vertex \( A, B, C, D \), the \( x \)-coordinate of the corresponding vertex in figure \( A'B'C'D' \) is the negative of the original \( x \)-coordinate. - This means if \( x \) is positive in figure \( ABCD \), it becomes negative in figure \( A'B'C'D' \), and vice versa. 2. **Comparison of \( y \)-coordinates:** - Similarly, for each vertex \( A, B, C, D \), the \( y \)-coordinate of the corresponding vertex in figure \( A'B'C'D' \) is also the negative of the original \( y \)-coordinate. - This means if \( y \) is positive in figure \( ABCD \), it becomes negative in figure \( A'B'C'D' \), and vice versa. ### Summary - The transformation results in each vertex being reflected through the origin, which means both the \( x \)- and \( y \)-coordinates change signs. - This reflection leads to a figure that is congruent to the original but located in the opposite quadrant of the coordinate plane. To visualize this, you would draw the original figure \( ABCD \) and then plot the new points \( A', B', C', D' \) based on the transformations described.