🔄 Flip, Turn & Slide! Mastering Geometric Transformations

Transformations are everywhere—from the symmetry of a butterfly’s wings to the rotation of a bicycle wheel. In the O Level syllabus, you must be able to **describe** and **perform** reflections, rotations, and translations with confidence. Mastery of these moves not only boosts your geometry marks but also strengthens spatial reasoning for coordinate geometry and vectors.

Core Concepts: Reflect 🔍, Rotate 🔁, Translate ➡️

**Reflection**: Flip a shape over a line called the mirror line. Every point and its image are the same perpendicular distance from the line, e.g. reflect over $y = x$ or the $x$ -axis.
**Rotation**: Turn a shape about a fixed point (the centre) through a given angle and direction (clockwise or anticlockwise). In matrices, a $90^{\circ}$ anticlockwise rotation about the origin is
$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$
.
**Translation**: Slide a shape without turning or flipping it. Described by a vector $\begin{pmatrix} a \\ b \end{pmatrix}$ meaning $a$ units right and $b$ units up.

Worked Example: Mapping Triangle ABC

Problem: Triangle

ABC

has vertices

A(1,2)

B(4,2)

, and

C(1,5)

. Perform the following successive transformations and state the final coordinates: 1. Reflect

\triangle ABC

in the line

y = x

. 2. Rotate the image

90^{\circ}

clockwise about the origin. 3. Translate the result by the vector

\begin{pmatrix} 3 \\ -2 \end{pmatrix}

Step 1 (Reflection): Swap each coordinate because the mirror line is $y = x$ .
$A'(2,1),\; B'(2,4),\; C'(5,1)$
Step 2 (Rotation): A $90^{\circ}$ clockwise turn multiplies each point by
$\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$
, giving
$A''(1,-2),\; B''(4,-2),\; C''(1,-5)$
Step 3 (Translation): Add $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ to every point. Final image
$A'''(4,-4),\; B'''(7,-4),\; C'''(4,-7)$

Consistent practice with tracing paper, grid diagrams, and vector notation will make these moves second nature. **Transform daily, perform confidently, and watch your geometry marks slide into the top band!**