A transformation moves a shape to a new position. The original is the object and the result is the image. In all three transformations here the image is congruent to the object: same shape, same size. Only its position and the way it faces change.
Congruent, from Unit 3, means same shape and same size. That one word tells you what a transformation is allowed to do: slide, turn or flip, but never stretch, shrink or bend. It also means you can always check your image by measuring it.
The three moves
| Name | Everyday word | What it does | To describe it fully you need |
|---|---|---|---|
| Translation | Slide | Moves every point the same distance in the same direction | How far across and how far up |
| Reflection | Flip | Produces a mirror image across a line | Where the mirror line is |
| Rotation | Turn | Turns the shape about a fixed point | The centre, the angle and the direction |
Saying a shape has been rotated is not enough, because there are infinitely many rotations. Saying it has been rotated 90 degrees clockwise about the point (2, 1) describes exactly one, and only that answer earns full marks.
Translation
Every point moves the same distance in the same direction, so the shape never turns and never flips. To translate by 3 across and 2 up, add 3 to every x-coordinate and 2 to every y-coordinate.
Reflection
Each point of the image is the same distance from the mirror line as the matching point of the object, on the opposite side. Any point sitting on the mirror line does not move at all, which is a useful check.
Your two hands are the same shape and size, but you cannot lay one on the other palm down. A reflection keeps every length and angle but reverses the sense, so a letter R becomes a backwards R. Sliding and turning never do this.
Rotation
A rotation turns the shape about a fixed point called the centre. Three things are needed: the centre, the angle, and the direction. A 180 degree turn is the same either way, but for 90 and 270 the direction certainly matters.
Trace the shape and the centre onto tracing paper, put your pencil point firmly on the centre, and turn the paper by the required angle. This is not a trick for weak students; it is what most people do, and it is reliable in a way that guessing is not.
A printed textile repeats one motif by translating it across the cloth. A rangoli or a temple carving is built by reflecting one section in several mirror lines. In animation and games, every moving object on screen is being translated and rotated many times a second.
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Triangle ABC has corners A(1, 1), B(4, 1), C(1, 3). Translate it 3 across and 2 up.
Reflect the point P(2, 3) in the y-axis, then in the x-axis.
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A transformation moves a shape. The original is the object, the result is the image.
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