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Understanding the definition of congruent?
As I understand it, two objects or figures are said to be congruent, if it is possible for one of them to match the other exactly, by means of rigid motions alone. Rigid motions being translation, rotation, and reflection (but not extension).
For 2-d figures these make sense, as rigid motions, as shown in the following annimation:
http://upload.wikimedia.org/wikipedia/commons/0/09...
Rigid motion meaning that you cannot adjust any of the points on the figure relative to any others. They all have to be moved as if the figure were rigid.
But what about 3-d solids?
Translation would still be considered a "rigid motion".
Rotation would still be considered a "rigid motion".
What about reflection?
In 2-d, reflection is considered a "rigid motion", because it really is rotation around a different axis than in-plane rotation.
In 3-d, there are no axes other than the 3 axes that are within the 3-dimensional space. You cannot make a mirror image of an object by rotating it alone.
Similar to these bundles of spheres. I can reflect one and make the other, but unless I can rotate through the 4th dimension, I cannot physically make this reflection of solids happen.
http://www.chemguide.co.uk/basicorg/isomerism/mode...
Another example is hands. Can I say that my hands are congruent?
1 Answer
- husoskiLv 78 years agoFavorite Answer
Yes, mirror reflections of asymmetric solids are congruent. The 2D reflection can't be done rigidly without leaving the plane, and the 3D reflection can be done rigidly in 4-space, but not without leaving the 3-space that the object starts and ends in.
Another way to look at this without the rotation is that two figures A and B are congruent iff there exists a 1:1 mapping f:A-->B such that ||X-Y|| = ||f(X)-f(Y)|| for all pairs of points X,Y in A. That is, the mapping preserves distances. Since preserving distances makes all triangles congruent, (easy SSS proof) then corresponding angles are all congruent.
That takes care of reflection without needing an extra dimension, or a precise definition of rigid motion.
