Are two congruent polygons always similar?
2 Answers
Yes. Congruent polygons are similar.
Explanation:
Start with the definition of congruence.
Congruent objects on a plane are those that can be transformed one into another by any combination of the following transformations:
- rotation around some point as a center,
- translation (shift) in certain direction,
- symmetry around some axis.
Continue with a definition of similarity.
Similar objects on a plane are those that can be transformed one into another by any combination of the following transformations:
- scaling using some point on a plane as a center and some real number as a factor of scaling,
- rotation around some point as a center,
- translation (shift) in certain direction,
- symmetry around some axis.
As you see, any pair of congruent objects are also similar since all transformations needed for congruence are included into similarity.
If you really want to add scaling (which is not necessary), choose any point on a plane as a center of scaling and factor of
Congruent shapes are always similar
Explanation:
The word 'congruent' means identical in all aspects..
It is the geometry equivalent of 'equal'.
Congruent figures have the same size, the same angles, the same sides and the same shape. They are IDENTICAL!
As a transformation we could say that the image maps onto the figure exactly.
The only things that can be different about congruent shapes are their position and orientation. (and the names of the vertices).
Similar figures have the same angles which means that they have the same shapes, but they differ in size. However, the sides are still in the the same ratio.
Congruent shapes are always similar , but similar shapes are usually not congruent - one is bigger and one is smaller.
In congruent shapes, the ratio of the corresponding sides is 1:1.
