Transformations

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infoWhy this? Reflections and enlargements deepen understanding of invariance, similarity, and coordinate change. Reasoning from mirror lines and centres of enlargement develops spatial precision and links geometric constructions with algebraic rules.

scheduleWhy now? This unit builds directly on Year 7 transformations and coordinate geometry, moving from basic grid methods to general horizontal, vertical, and diagonal reflections and centred enlargements. It prepares us to describe and combine transformations with greater formality.

A reflection is a transformation that produces a mirror image and preserves lengths and angles (it is an isometry).
In a reflection, the line of reflection is the perpendicular bisector of the segment joining any point and its image.
Reflecting in a horizontal line `y=b` keeps the `x`-coordinate the same and maps `y` to `y' = 2b - y`.
Reflecting in the x-axis is the special case `y=0`, which maps `(x,y)` to `(x,-y)`.
Reflecting in a vertical line `x=a` keeps the `y`-coordinate the same and maps `x` to `x' = 2a - x`.
Reflecting in the y-axis is the special case `x=0`, which maps `(x,y)` to `(-x,y)`.
Reflecting in the line `y=x` swaps the coordinates, so `(x,y)` maps to `(y,x)`.
Reflecting in the line `y=-x` swaps the coordinates and changes their signs, so `(x,y)` maps to `(-y,-x)`.
A point on the line of reflection stays in the same place after reflection (it is invariant).
An enlargement is a transformation that makes a shape similar to the original by scaling all distances from a centre of enlargement by the same factor.
The centre of enlargement is a fixed point: it does not move under the enlargement.
For scale factor `k`, every distance from the centre is multiplied by `|k|`, so `k>1` gives an enlargement and `0<k<1` gives a reduction.
A positive scale factor keeps the shape’s orientation the same, while a negative scale factor reverses the shape through the centre as well as scaling it.
Under enlargement, corresponding sides remain parallel and corresponding angles stay equal because the shapes are similar.
With centre `(a,b)` and scale factor `k`, a point `(x,y)` maps to `(x',y')` where `x' = a + k(x-a)` and `y' = b + k(y-b)`.
On a coordinate grid, the image point for an enlargement lies on the straight line through the centre and the original point, at `k` times the directed distance from the centre.

rocket_launchYou must be able to

Reflect a point across a horizontal line `y=b` by keeping `x` the same and calculating `y' = 2b - y`, then plot the image point accurately on the grid.
Reflect a point across a vertical line `x=a` by keeping `y` the same and calculating `x' = 2a - x`, then plot the image point accurately on the grid.
Reflect a point in `y=x` by swapping the coordinates to get `(y,x)`, and reflect a point in `y=-x` by mapping `(x,y)` to `(-y,-x)`.
Reflect a 2D object by reflecting each vertex (corner) and joining the image vertices in the same order to form the reflected shape.
Check a reflection is correct by showing each original point and its image are the same perpendicular distance from the line of reflection.
Enlarge a point from a given centre by finding the vector from the centre to the point and multiplying that vector by the scale factor, then adding it back to the centre.
Enlarge a 2D object from a given centre by enlarging each vertex from the centre and joining the image vertices in the same order to form the enlarged shape.
Verify an enlargement by checking that corresponding side lengths scale by `|k|`, angles match, and corresponding sides are parallel.

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