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:
1. Reflection: Reflect the shaded triangle across the y-axis to create a mirrored image on the left side of the square (Figure II).
2. Rotation: Rotate the shaded triangle 90 degrees counterclockwise around the origin to create a new triangle aligned with the positive x-axis (Figure III).
3. Translation: Move the newly rotated triangle to the top of the square by translating it vertically upward (Figure IV).
4. Reflection: Reflect the newly placed triangle across the x-axis to create a mirrored image at the bottom of the square (Figure V).
5. Rotation: Rotate the newly reflected triangle 180 degrees around the origin to create a new triangle aligned with the negative y-axis (Figure VI).
6. Translation: Move the newly rotated triangle to the left side of the square by translating it horizontally to the left (Figure VII).
7. Reflection: Reflect the newly placed triangle across the y-axis to create a mirrored image on the right side of the square (Figure VIII).
After performing these transformations, the entire square design should be complete.
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To create the rest of the square design using the shaded triangle, we will need to perform a combination of translations, reflections, and rotations. Here's a step-by-step explanation of the transformations needed:
1. Translation: First, we need to translate the shaded triangle horizontally so that it is located on the positive x-axis. To do this, we move the shaded triangle to the right by the same distance as the length of one side of the square.
2. Reflection: Once the shaded triangle is positioned on the positive x-axis, we need to reflect it to create another triangle on the negative x-axis. This reflection is achieved by flipping the shaded triangle over the y-axis.
3. Rotation: Now we have two triangles on the positive and negative x-axes. To create the remaining two triangles, we need to rotate these triangles clockwise by 90 degrees. This rotation will move the positive x-axis triangle to the positive y-axis, and the negative x-axis triangle to the negative y-axis.
4. Reflection: Finally, we need to reflect the positive y-axis triangle to create the triangle on the negative y-axis. This reflection is achieved by flipping the positive y-axis triangle over the x-axis.
By following these steps, we have now created four congruent triangles inside the square, forming the square design. Each triangle is a transformation of the original shaded triangle, where translations, reflections, and rotations were used to produce the other triangles in the design.