Two Transformations, One Chain

A shape can finish in a completely different place after just two moves. But here is the twist... swapping those moves can change the final answer. So how do you keep control? Treat transformations like instructions in a queue: first one, then the next.

Start with the original shape on a coordinate grid. Read every point across first for x, then up or down for y. Horizontal BEFORE vertical. If you reverse that order, you plot a different point before the transformations even begin. Now read the two instructions from top to bottom. Complete the first transformation for ALL the vertices, and label the new image. Only then use that image as the starting shape for instruction two. The first answer becomes your new input. A translation slides every point by the same amount. A rotation turns every point around its stated centre. Watch the corner order and the shape's orientation: a turn does not make a mirror image. It stays the same shape, simply facing another way.

Let's work through an example. Triangle A B C has vertices A at one, one; B at three, one; and C at one, two. First, translate it four squares right and one square up. Then rotate the image ninety degrees clockwise about the origin. Find the final coordinates. Do instruction one. Add four to each x-coordinate and one to each y-coordinate. A moves to five, two. B moves to seven, two. C moves to five, three. Pause there... these three points form the translated image, ready for the rotation. For a ninety-degree clockwise turn about the origin, each point at x, y moves to y, negative x. Apply that to the translated image, not the original. A finishes at two, negative five. B finishes at two, negative seven. C finishes at three, negative five. Those are the final coordinates.

Remember: read x before y, follow the instructions in order, and make each new image your next starting shape. Two transformations, one careful chain... you've got it!