Look at the shaded triangle in the drawing below. Describe in detail the transformations needed to create the rest of the square design. You can use a combination of translations, reflections, and rotations of the shaded triangle (Figure I) to create the square design.
A square is drawn on a coordinate plane with the center of the square on the origin. Two diagonals are drawn through the origin forming four congruent triangles on the inside of the square. The triangle that overlaps the positive y-axis is shaded.
To create the rest of the square design, we can use the following transformations on the shaded triangle:
1. Reflect the shaded triangle across the y-axis. This will create another shaded triangle on the negative y-axis.
2. Rotate the shaded triangle by 90 degrees clockwise around the origin. This will create another shaded triangle on the negative x-axis.
3. Reflect the shaded triangle across the x-axis. This will create another shaded triangle on the positive x-axis.
4. Rotate the shaded triangle by 180 degrees around the origin. This will create another shaded triangle on the negative y-axis, overlapping the original shaded triangle.
5. Reflect one of the shaded triangles (either the original or the one overlapping it) across the line y=x. This will create another shaded triangle in the upper right quadrant of the square.
6. Reflect one of the shaded triangles (either the original or the one overlapping it) across the line y=-x. This will create another shaded triangle in the upper left quadrant of the square.
After performing these transformations, we will have six shaded triangles that together form the square design inside the larger square.