Positions and Transformations

Coordinates describe an exact position on a grid. They are written as a pair in brackets, $(x, y)$.

The grid is built from two number lines, called axes. The horizontal one is the x-axis; the vertical one is the y-axis. They cross at the origin, the point $(0, 0)$.

The golden rule for reading a coordinate is along the corridor, then up the stairs — the x-value first, then the y-value.

The two axes split the grid into four regions called quadrants. Coordinates can be negative:

• A negative $x$ means go to the left of the origin.
• A negative $y$ means go down from the origin.

So $(3, 2)$ is 3 right and 2 up, while $(-2, -3)$ is 2 left and 3 down.

A reflection flips a shape across a mirror line, like the reflection you see in a mirror. The new shape is called the image.

The key facts about a reflection:

• The image is the same size as the original — reflection never stretches or shrinks.
• Every point of the image is the same distance from the mirror line as the matching point on the original.
• The image is flipped — it faces the opposite way, like a left hand reflecting to a right hand.

To reflect a shape, take each corner one at a time, count its distance to the mirror line, then count the same distance on the other side. Join the new corners up.

Mirror lines are often the axes or simple lines like $x = 2$ or $y = -1$. A point on the mirror line does not move at all.

A rotation turns a shape around a fixed point, called the centre of rotation. To describe a rotation fully you need three things: the centre, the angle of turn, and the direction (clockwise or anticlockwise).

Common angles are quarter, half and three-quarter turns:

• A quarter turn is $90\degree$.
• A half turn is $180\degree$ (clockwise and anticlockwise look the same here).
• A three-quarter turn is $270\degree$.

Like a reflection, a rotation keeps the image exactly the same size and shape — it just sits in a new position, turned around. A piece of tracing paper helps: trace the shape, hold a pencil on the centre, and turn the paper.

The point at the centre of rotation is the only point that does not move.

A translation simply slides a shape from one place to another. The shape does not turn and it does not flip — every point moves the same distance in the same direction.

To describe a translation, give two moves:

• how far left or right, and
• how far up or down.

For example, "5 squares right and 2 squares down" fully describes the slide in the diagram.

The image is congruent to the original — exactly the same size, shape and the same way up. Translation is the gentlest of the three transformations: nothing about the shape itself changes, only where it sits.

A neat way to check your work: pick one corner, count the slide, and make sure every other corner moves by exactly the same amount.

A shape has symmetry when it looks the same after a reflection or a rotation. There are two kinds to know.

Line symmetry (also called reflective symmetry) means a shape has a line of symmetry — a mirror line that splits it into two matching halves. A square has 4 lines of symmetry; a rectangle has 2; an equilateral triangle has 3.

Rotational symmetry means a shape looks the same as you turn it through a full $360\degree$. The order of rotational symmetry is how many times it matches itself in one full turn. A square has rotational symmetry of order 4; an equilateral triangle has order 3.

Every shape has rotational symmetry of at least order 1 (it always matches after a full turn) — but we only say it has rotational symmetry if the order is 2 or more.

Getting confident with positions and transformations sets you up well:

• Graphs and algebra — plotting straight-line graphs uses coordinates constantly.
• More transformations — later you will meet enlargement, which changes a shape's size as well as its position.
• Geometry reasoning — congruence and symmetry help you prove shapes are identical.
• In everyday life, coordinates and transformations appear in maps, computer graphics, art, design and games.

If a later geometry or graphing topic feels tricky, it is often a coordinate or transformation skill underneath. Keep this guide handy — plotting and moving shapes gets quicker every time you practise.