Identifying a Transformation From Its Image

Every transformation leaves fingerprints in the coordinates. Pick one vertex and its image, write both as ordered pairs, and ask what rule turns the first into the second. If x and y both shift by the same fixed amount for every point, that's a translation, sliding the whole figure without turning or flipping it. If the coordinates swap places, like (2,3) going to (3,2), that's a reflection across the line y = x. If a coordinate swaps and also changes sign, that points to a 90 degree rotation about the origin. If every coordinate just scales by the same factor from a fixed center, that's a dilation, which changes size but not shape.

The trick is to check more than one point, because a single pair of coordinates can sometimes match more than one type of move. Confirming the same rule holds for a second vertex, as with (5,1) going to (1,5) also swapping coordinates, locks in the answer: both points had x and y traded with no sign change, so the whole figure reflected across y = x rather than rotating or translating.

This skill connects directly to orientation and rigid motion ideas nearby in the unit. Translations, rotations, and dilations all preserve the order in which you read a figure's vertices, clockwise or counterclockwise. Reflections are the odd one out: they flip that order every time. That single fact becomes a fast check whenever coordinates alone feel ambiguous, and it carries forward into proving two figures are congruent or similar based on which transformations map one onto the other.