Problem 4-6
4-6 A
4-6 A
A translation is equivalent to the two reflections over parallel lines. In the case above, the original object is translated a distance of 13.46 units, 12.04 units to the right and 6.02 units up. As we move the parallel lines and the object to any spot (the object doesn't just have to be to the left of the first line but can be anywhere). No matter where we move the object, the translation stays the same. Only when we move the parallel lines, changing their slope or distance, is when we change the translation that is happening. After looking closer, it holds for wherever the lines or shapes are that the distance between the parallel lines doubled is also the distance between the first and last shapes.
This makes sense if we logically think it through, as we know the distance from the first shape to the second it just double the distance from the second shape to the first parallel line. Similarly, the distance from the second to the third shape is double the distance from the second shape to the second parallel line. So if we put this all together the distance from the first to the third shape is double the distance between the lines.
4-6 B
4-6 B
A rotation is equivalent to two reflections of intersecting lines. In the case above the original shape is rotated 96.19 degrees. The rotation holds for any angle of intersection and placement of the shape. Note that when you move the shape around but do not change the intersection angle of the lines, the rotation angle stays the same no matter where the shape is. So the rotation angle is based solely on the angle of intersection. Changing around the angle of intersection, we see that the rotation angle is always twice the angle of intersection. Note, once the angle of intersection is above 180 degrees, it's not exactly double as Geogebra starts the degrees back to 0 at 360. So if we say the angle of intersection is 181, then instead of saying the rotation angle is 362 degrees, it says its 2 degrees, those this is technically the same angle.
We can think of why it is doubled using the same logic as what we did above. Between the first and second shape is just double that angle from the middle shape to that first line. And between the second and third shape is just double the angle from the middle shape to the second line. So in the end from the first to the third shape is just double the angle of the intersecting lines.