explain why the image of a translation is congruent to it's pre image
Sure! To understand why the image of a translation is congruent to its preimage, let's first define what a translation is.
A translation is a type of transformation in geometry that moves every point of an object in a straight line without changing its shape or orientation. It involves shifting all the points of an object by the same distance and direction.
Consider an object, let's say a polygon, on a coordinate plane. The preimage of the polygon is its original position or shape before any transformation is applied. The image of the polygon is the new position or shape after the translation is applied.
Now, let's see why the image of a translation is congruent to its preimage:
1. Distance: When we perform a translation, we shift every point in the object by the same distance and direction. This means that all the corresponding points in the preimage and the image have the same distance between them.
2. Parallelism: A translation does not alter the parallelism of the object. This means that if any two sides of the preimage are parallel, their corresponding sides in the image will also be parallel. Likewise, if any two angles in the preimage are equal, their corresponding angles in the image will also be equal.
3. Shape: Since a translation preserves distances and parallelism, the overall shape of the object remains the same. This means that the preimage and the image will have the same number of sides, vertices, and angles.
Based on these properties, we can conclude that the image of a translation is congruent to its preimage because both the shape and size of the object are preserved during the transformation. In other words, the translated object is an exact copy of the original object, just shifted to a different location.