Answer:
- true
- true
- false
- false
Step-by-step explanation:
One characteristic of a reflection that is useful for answering this question is that it always reverses the clockwise/counterclockwise orientation of a figure. Rotation and translation have no effect on that orientation.
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1. translation by reflection
Reflection across two parallel lines has the net effect of translating a figure twice the distance between the parallel lines. So, one way to effect a translation using two lines of reflection is to draw one of them through the perpendicular bisector of a point and its translated image. Then the other line would be drawn parallel to the first through the image point.
True: translation can be replaced by two reflections
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2. translation by rotation
A single point can be moved from one set of coordinates to another by rotating around any point on the perpendicular bisector of the original and its image. To "undo" the change in direction of other points in the image, the image can be rotated an equal angle in the reverse direction about the point that is in its proper place.
That is, if we rotate figure ABC an amount of X° about a point on the perpendicular bisector of AA', so that A ends up at A', then the translation can be finished by rotating that figure by -X° about point A'.
The simplest case is an initial rotation of 180° about the midpoint of AA', followed by another rotation of 180° about A'.
True: translation can be replaced by two rotations
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3. rotation by reflection
As discussed above, reflection changes orientation and rotation does not.
However, a rotation can be replaced by <em>two</em> reflections. The rotation angle is equal to twice the angle between the lines of reflection. The point where the lines of reflection meet is the center of rotation.
False: rotation can be replaced by reflection
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4. reflection by rotation and translation
As discussed above, reflection changes orientation, but rotation and translation do not.
False: reflection can be replaced by rotation and translation
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