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Segment is the perpendicular bisector of segment . Find each transformation mentally.
If two figures are congruent, we can always find a rigid transformation that takes one onto the other.
Look at congruent figures and . It looks like might be a translation, rotation, and reflection of . But is there a way to describe a sequence of transformations without guessing where the line of reflection might be?
Our goal is to take onto . Then we want to take the image of onto without moving and . Finally, we need to take the image of onto without moving any of the matching points.
We can start with translation: Translate triangle by the directed line segment from to .
Now, a pair of corresponding points coincides. Is there a transformation we could use to take onto that leaves and in place? Rotations have a fixed point, so rotate triangle by angle using point as the center.
Now, two pairs of corresponding points coincide. Reflecting across line will take onto , which is what we were trying to do. We know and won’t move since points on the line of reflection don't move. How do we know will end up on ? Since the triangles are congruent, and are the same distance from the line of reflection.
It is always possible to describe transformations using existing points, angles, and segments. It could take an extra step, but we can be confident that transformations work if we don't guess where the line of reflection or center of rotation might be.