🔄Transformations

Figures move too: an elevator rising as one block, the image in a mirror, a spinning pinwheel, an enlarged photo. Geometry calls these four actions transformations. A transformation changes only a figure's position or size — the shape itself stays — and that is exactly what makes it useful.

The four basic moves

• Translation: the whole figure slides, keeping its direction;
• Reflection: the figure flips over a line, like a mirror;
• Rotation: the figure turns about a point through an angle;
• Enlargement or reduction: the figure stretches or shrinks by a factor — the similarity scaling from congruence & similarity.

Translation and reflection

In Cartesian coordinates every move can be written as arithmetic on coordinates. Translation: to move the point $(2,3)$ right 3 and down 1, add 3 to the x-coordinate and subtract 1 from the y-coordinate, landing at $(5,2)$.

Reflection: flipping over the x-axis flips the sign of the y-coordinate — $(2,3)$ becomes $(2,-3)$; over the y-axis, the sign of the x-coordinate flips — $(2,3)$ becomes $(-2,3)$. The left-right swap you see in a mirror is a reflection.

Rotation and enlargement

Rotation: turning $180^\circ$ about the origin flips both signs, so $(2,3)$ lands at $(-2,-3)$; a $90^\circ$ turn swaps the two coordinates and flips one sign. Change the centre or the angle and the landing spot changes too.

Enlargement: scaling by a factor of 2 about the origin multiplies every coordinate by 2, sending $(2,3)$ to $(4,6)$. A factor above 1 grows the figure, below 1 shrinks it — the shape never changes.

x3y-2

(3, -2) → Quadrant IV

Walk x steps horizontally, then y steps vertically — the ordered pair (x, y) pins down exactly one point.

Set out a triangle on the board, say $(0,0)$, $(4,0)$, $(4,3)$; then add $(2,1)$ to every pair of coordinates and place the second triangle — the whole figure has translated up and to the right. Changing coordinates is how you steer a figure.

Symmetry is reflection onto yourself

A symmetric figure coincides with itself after a reflection across its axis — symmetry is the property of "unchanged by reflection"; a pinwheel's blades coincide with themselves after a certain turn, which is rotational symmetry. Transformations are more than ways to move figures — they are a mirror for seeing what a figure really is.

Check yourself

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Quick quiz

↺ Try again

1. 1. Where does the point (2,3) land after a reflection over the x-axis?

   (-2,3)(2,-3)(3,2)

2. 2. The point (2,3) is translated right 3 and down 1. Where does it land?

   (5,2)(5,4)(-1,2)

3. 3. A figure is enlarged by a factor of 2. What changes and what does not?

   Size changes, shape does notShape changes, size does notBoth change