Position and Movements

A coordinate point is written $(x, y)$. The first number tells you how far to go across from the origin (positive right, negative left), and the second tells you how far to go up or down (positive up, negative down).

The two axes split the grid into four quadrants:

• Top right: both positive.
• Top left: $x$ negative, $y$ positive.
• Bottom left: both negative.
• Bottom right: $x$ positive, $y$ negative.

A neat trick for the midpoint of a line segment is to average the coordinates: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$. The midpoint of $(2, 6)$ and $(8, 2)$ is $(5, 4)$.

A reflection flips a shape across a mirror line. Every point on the original shape (the object) has a matching point on the new shape (the image), and the two are the same perpendicular distance from the mirror line.

The mirror line can be horizontal (like $y = 2$), vertical (like $x = -1$), or slanted (like $y = x$).

A reflection does not change the size or shape of the figure — only its position is flipped. The image is congruent to the object.

To describe a reflection in full you need two things: the word "reflection", and the equation of the mirror line. For example: "Reflection in the line $y = 1$."

A rotation turns a shape about a fixed point called the centre of rotation. To describe a rotation you need three things:

• the centre of rotation (a coordinate point),
• the angle of rotation ($90\degree$, $180\degree$, or $270\degree$),
• the direction: clockwise or anticlockwise. (A $180\degree$ turn is the same either way.)

Tracing paper makes a rotation easy. Trace the object, push a pin through the centre, then turn the paper by the given angle and direction.

Rotations also preserve size and shape — only orientation changes. The image is congruent to the object.

A translation slides a shape across the grid. There is no flip and no turn — every point moves the same distance in the same direction.

A translation is described by a column vector: $\begin{pmatrix} a \\ b \end{pmatrix}$ The top number $a$ is the horizontal move (positive right, negative left). The bottom number $b$ is the vertical move (positive up, negative down).

For example, $\begin{pmatrix} 4 \\ -2 \end{pmatrix}$ means 4 right and 2 down. Translations preserve size, shape and orientation — only the position changes.

An enlargement changes the size of a shape. To describe one you need:

• the centre of enlargement (a coordinate point),
• the scale factor (the number that every length is multiplied by).

With scale factor $2$, every length doubles. With scale factor $3$, every length triples.

Each point of the image sits scale-factor times further from the centre than the matching point on the object. So a point that was $3$ squares from the centre will be $6$ squares from the centre after a scale-factor-$2$ enlargement.

Enlargements change size but keep shape — the image is similar to the object (same shape, different size).

Symmetry describes the patterns hidden inside a shape.

A line of symmetry is a straight line that splits a shape into two mirror images. If you folded the shape along the line, the two halves would match perfectly.

• A square has 4 lines of symmetry.
• A rectangle (that is not a square) has 2.
• A regular polygon with $n$ sides has $n$ lines of symmetry.

Rotational symmetry counts how many times a shape lands back on itself in one full turn. This count is the order of rotational symmetry.

• A square has rotational symmetry of order $4$ (it lands on itself every $90\degree$).
• A regular hexagon has order $6$.
• A shape with no rotational symmetry has order $1$ (it only matches once, after a full $360\degree$ turn).

A useful pattern: in any regular polygon, the number of sides, the number of lines of symmetry, and the order of rotational symmetry are all the same number.

Confidence with coordinates and transformations sets you up well for what comes next:

• Algebra and graphs plot equations as lines and curves on a coordinate grid.
• Later geometry builds combined transformations and inverse transformations.
• Pythagoras' theorem uses coordinates to find the distance between two points.
• Outside the classroom these ideas drive computer graphics, animation, robotics and architecture.

If a later geometry topic feels tricky, the fix is often to draw the picture on a grid and step through it carefully. Keep this revision close to hand.