Slide, flip, turn, scale
Press a button to move the triangle. The grey copy is where it was, the blue one is where it is now, and the readout gives the rule that sends every point (x, y) to its new place, with the three corners worked out. Check one corner by hand before you trust the rule. Then compare side lengths: the slides, flips and turns keep them, and only the scale changes them. Drag any corner to start from a different triangle.
What's going on
Each button is a rule for coordinates. A slide (translation) adds the same amounts to every x and every y: (x, y) → (x + dx, y + dy). A flip (reflection) in the x-axis changes the sign of y; in the y-axis it changes the sign of x; in the line y = x it swaps them. A quarter turn anticlockwise about the origin sends (x, y) to (−y, x): check it on (1, 0), which must land at (0, 1). Doing that twice gives the half turn (x, y) → (−x, −y).
Slides, flips and turns are rigid motions: they move the triangle without stretching it, so every side keeps its length and every angle keeps its size. The new triangle is congruent to the old one. A flip does one extra thing a slide or turn cannot: it turns the triangle over, so reading the corners A, B, C now goes round the other way. No sequence of slides and turns can undo that.
The scale (dilation) multiplies both coordinates by k. Lengths become |k| times as long (lengths are never negative), angles stay the same, and the area becomes k² times as big. For k ≠ 0 the result is similar to the original but not congruent, unless k = 1 or −1. A negative k also includes a half turn. (k = 0 would squash everything onto the origin.) Try k = −1 and compare it with the half turn.
Sheets: Slide, flip, turn, scale. What comes later: the shift, stretch, flip demo for graphs and the matrix demo, which writes these same rules as 2 × 2 matrices.