Two triangles can look completely different on a page — one rotated, one flipped, one sitting somewhere else entirely — and still be, in a precise mathematical sense, exactly the same shape. Geometric transformations formalise the small set of moves that can turn one shape into another while preserving (or, in one case, deliberately changing) its size, letting geometry classify shapes by family rather than by their arbitrary position on a page.
Three moves that never change size or shape at all
A translation slides a shape to a new position without rotating, flipping, or resizing it — every point moves the same distance in the same direction. A rotation turns a shape around a fixed point by some angle, again without changing its size or its internal proportions. A reflection flips a shape across a line, producing its mirror image — same size, same shape, but with its orientation reversed, which is why a reflected shape can't always be slid or rotated back onto the original without lifting it off the page; some shapes, and famously human hands, are "chiral," meaning no combination of sliding and rotating alone can turn the left version into the right one, only a reflection can. Translations, rotations, and reflections are all "rigid" transformations, called isometries — they preserve every distance and every angle in the original shape exactly.
The fourth move, and why it breaks the pattern deliberately
A dilation, unlike the other three, deliberately changes a shape's size — scaling every point's distance from a fixed centre by the same factor, which shrinks or enlarges the shape while keeping its proportions and angles identical. Two shapes related by a dilation (possibly combined with the other three moves) are called similar rather than congruent — same shape, same angles, but different overall size. Together, these four transformations, and combinations of them, are enough to describe the relationship between any two shapes that share the same fundamental form: two shapes are congruent if one can be produced from the other using only translations, rotations, and reflections, and similar if a dilation is also needed.
What we're still unsure about
The classification of these four transformations, and their formal properties, is fully settled classical geometry — this isn't a subject of live mathematical dispute. Where the concept gets genuinely richer is in how far it generalises: the same basic idea of transformations that preserve or systematically change certain properties extends into far more abstract mathematics, including group theory and the study of symmetry in physics, where the question of exactly which transformations a given physical law or structure remains unchanged under (its "symmetry group") is often a deep, actively researched question rather than something read off a shape by eye, the way it can be with a simple two-dimensional figure.
This sits inside Geometric Transformations, one of eight topics in Geometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.