Examples 1 and 2 1. Triangle \( X Y Z \) has vertices \( X(0,2), Y(4,4) \), and \( Z(3,-1) \). Graph \( \triangle X Y Z \) and its image after a rotation of \( 180^{\circ} \) counterclockwise about \( Z \).

The rotation of triangle \( XY Z \) by \( 180^{\circ} \) counterclockwise about point \( Z(3,-1) \) involves finding the image of points \( X(0,2) \) and \( Y(4,4) \). To perform this rotation, subtract the coordinates of \( Z \) from those of \( X \) and \( Y \), then apply the 180-degree transformation, which negates both the x and y displacements, and finally add the coordinates of \( Z \) back. For point \( X(0,2) \): - Displacement from \( Z \): \( (0-3, 2+1) = (-3, 3) \) - After rotation: \( (3, -1) + (3, -3) = (6, -4) \) For point \( Y(4,4) \): - Displacement from \( Z \): \( (4-3, 4+1) = (1, 5) \) - After rotation: \( (3, -1) + (-1, -5) = (2, -6) \) So the new coordinates are: - \( X' (6, -4) \) - \( Y' (2, -6) \) Now, graph \( \triangle XYZ \) using the original points \( (0,2) \), \( (4,4) \), and \( (3,-1) \), and then plot the rotated triangle \( X' (6, -4) \), \( Y' (2, -6) \), and \( Z(3,-1) \). Draw lines to connect these points to visualize both triangles on the graph. Enjoy the geometric dance!