Why can you treat a transformation like a machine for points?
You already know a function as something that takes an input and gives exactly one output. A transformation does that too, except your input is usually a point or a whole shape on a grid. If you feed in the original shape, each point lands in one new place, so you get one new shape out.
A geometric transformation is a rule that sends every point of your starting figure to a new point. Your starting figure is the pre-image, and the result after the rule is applied is the image.
You can write this idea as image = T(pre-image), where T names the rule. That rule could be a slide, a turn, a flip, or a resize. The useful idea is that a whole shape is just lots of points being transformed one by one.
If you transform a triangle, you are not moving “the triangle” as one object in some magical way. You are moving each vertex and every point on its sides according to the same rule, and that builds the image.
How do you know whether a transformation keeps the shape exactly the same?
Here is the big split: some transformations only move a figure around, while others change its size. If lengths and angles stay unchanged, your image matches the original exactly in shape and size. That means the two figures are congruent.
An isometric transformation keeps all distances the same, so your figure keeps both its shape and its size.
The three transformations you meet most often in this group are translations, rotations, and reflections. A surprising detail: a reflection does keep all lengths and angles, even though the figure looks “backwards” afterwards.
Figures are congruent when they have the same shape and the same size. Figures are similar when their matching angles are equal and their matching side lengths all have the same ratio.
If your transformation is isometric, the image is congruent to the pre-image. If the figure is resized by a dilation, the image is usually only similar, not congruent.
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