Unit 19: Moving Shapes

A shift moves everything the same way

A translation slides a shape without turning it. Every point moves by the same amount.

Move 3 right and 2 up, and the point (4, 1) goes to (7, 3). Add 3 to the first number, add 2 to the second.

The shape does not change at all — only where it is.

A flip swaps a sign

A reflection flips a shape over a line, like a mirror.

Flip over the across-line and the second number changes sign: (3, 4) goes to (3, -4).

Flip over the up-line and the first number changes sign instead: (3, 4) goes to (-3, 4).

One mirror, one sign.

A quarter turn swaps and flips

Turn a point 90° anticlockwise about the origin and (x, y) becomes (-y, x).

So (3, 1) goes to (-1, 3). Swap the two numbers, then change the sign of the new first one.

Half a turn is easier still: (x, y) becomes (-x, -y). Both signs flip.

Rotational symmetry

A shape has rotational symmetry if turning it lands it back on itself before a full turn.

A square looks the same after every 90°, so it has order 4. An equilateral triangle has order 3. A regular polygon with n sides has order n.

Divide 360 by the order and you get the smallest turn that works.

Reflecting in the diagonal

Flipping over the across-line or the up-line changes one sign. Flipping over the diagonal line y = x does something different: it swaps the two numbers.

(3, 7) becomes (7, 3).

That makes sense — the diagonal treats across and up identically, so a reflection in it must trade one for the other.

A point on the mirror does not move

Reflect (5, 5) in the line y = x and you get (5, 5) again. It was already on the mirror.

Every reflection has a whole line of points that stay put — the mirror itself.

A rotation has just one such point, the centre. A translation has none at all, which is why nothing about a shape can be used to find where it came from.

Two moves make a third

Do a transformation, then another. The overall effect is always one transformation — often a different kind.

Two reflections in parallel mirrors give a translation, twice the gap between the mirrors.

Two reflections in crossing mirrors give a rotation, twice the angle between them. That is exactly how a kaleidoscope works.

Order can matter

Translate then rotate, or rotate then translate? Those usually give different answers.

Try it with a shift of 3 right and a quarter turn. Shifting first moves the shape away from the centre before it swings; turning first swings it and then moves it.

So transformations behave like the made-up operations in unit 23 — not always commutative, and worth checking rather than assuming.

Working out the move from before and after

Given a shape and its image, which transformation was used?

Same size, same way up — a translation. Just read off how far it went.
Same size, mirror image — a reflection. The mirror is halfway between matching points.
Same size, turned — a rotation.
Different size — an enlargement.

Check the size first. It rules out three of the four immediately.

Finding the mirror line

If (1, 4) reflects to (1, 10), the mirror must be exactly halfway between them.

That is the line y = 7, since (4 + 10)/2 = 7.

So finding a mirror line is finding a midpoint — the same tool from unit 16, put to a new use.

The centre stays still

An enlargement needs two things: a scale factor and a centre.

Every point moves along the line from the centre, ending up k times as far out.

The centre itself never moves — it is the projector lamp, and everything else is the picture growing on the wall.

A negative scale factor turns it upside down

A scale factor of -2 sends every point to the opposite side of the centre, twice as far out.

The shape ends up twice as big and upside down — which is exactly a rotation of 180° combined with an enlargement.

It is also how a pinhole camera works, and why the image inside one is inverted.

An enlargement multiplies

An enlargement from the origin multiplies both numbers by the scale factor.

Scale factor 3 sends (2, 4) to (6, 12). Everything gets three times as far from the origin.

A factor between 0 and 1 squeezes it instead — factor 1/2 sends (2, 4) to (1, 2).

What survives, and what does not

Shifts, turns and flips keep everything: lengths, angles and area. Shapes stay congruent.

Enlargements keep the angles but change the lengths — shapes stay similar, and the area changes by k^2, exactly as in unit 15.

A stretch in one direction only keeps neither. It changes the shape.