Everything covered in this lesson, in one place - useful for revision or printing.
\( \text{object} \rightarrow \text{transformation} \rightarrow \text{image} \)
A transformation moves or resizes a shape on the grid. The original shape is the object; the result is the image.
In this lesson you will learn to:
\( \text{translate} \quad \text{reflect} \quad \text{rotate} \quad \text{enlarge} \)
The four transformations you must know:
\( \text{rigid} = \text{same size and shape} \)
Translations, reflections and rotations are rigid: the image is congruent to the object — same side lengths, same angles.
\( A \rightarrow A' \qquad B \rightarrow B' \)
Image points get a prime mark: point \(A\) maps to \(A'\) (read “A prime”).
\( \text{slide, flip, turn, resize} \)
Key idea: three transformations keep the shape congruent (translation, reflection, rotation); the fourth changes its size (enlargement).
\( (x;\ y) \rightarrow (x+a;\ y+b) \)
A translation slides every point the SAME distance in the SAME direction.
\( P(2;\ 3) \rightarrow P'(5;\ 1) \)
Translate \(P(2; 3)\) by “3 right, 2 down”:
\( A(1;\ 4) \rightarrow A'(-2;\ 6) \)
To find the rule, subtract: image minus object.
\( (x;\ y) \rightarrow (x+a;\ y+b) \)
Key idea: right/left changes \(x\); up/down changes \(y\). Every point of the shape moves by exactly the same amounts.
\( (x;\ y) \rightarrow (x;\ -y) \)
Reflecting in the x-axis flips the shape vertically: \(x\) stays, \(y\) changes sign.
\( (x;\ y) \rightarrow (-x;\ y) \)
Reflecting in the y-axis flips the shape horizontally: \(y\) stays, \(x\) changes sign.
\( (x;\ y) \rightarrow (y;\ x) \)
Reflecting in the line \(y=x\) swaps the coordinates.
\( \text{object and image are the same distance from the mirror} \)
A reflection works like a mirror:
\( x\text{-axis}: (x; -y) \quad y\text{-axis}: (-x; y) \quad y=x: (y; x) \)
Key idea: memorise the three coordinate rules — they turn any reflection question into simple sign changes or a swap.
\( 90^\circ \quad 180^\circ \quad 270^\circ \)
A rotation turns the shape about a fixed point (for Grade 8, usually the origin). You need the angle and the direction (clockwise or anticlockwise).
\( (x;\ y) \rightarrow (-x;\ -y) \)
Rotating \(180^\circ\) about the origin changes BOTH signs.
\( 90^\circ \text{ anticlockwise}: (x;\ y) \rightarrow (-y;\ x) \)
For \(90^\circ\) about the origin:
\( 180^\circ: (-x; -y) \qquad 90^\circ\text{ acw}: (-y; x) \qquad 90^\circ\text{ cw}: (y; -x) \)
Key idea: rotations about the origin have coordinate rules too. The shape stays congruent — only its orientation changes.
\( (x;\ y) \rightarrow (kx;\ ky) \)
An enlargement from the origin multiplies BOTH coordinates by the scale factor \(k\).
\( \text{new length} = k \times \text{old length} \)
Every side of the image is \(k\) times the matching side of the object.
\( k = \dfrac{\text{image length}}{\text{object length}} \)
To find the scale factor, divide a pair of matching lengths:
\( (x;\ y) \rightarrow (kx;\ ky) \)
Key idea: multiply coordinates and lengths by \(k\); angles stay the same. \(k>1\) grows the shape, \(k<1\) shrinks it.
\( \text{new perimeter} = k \times \text{old perimeter} \)
Since every side is multiplied by \(k\), the whole perimeter is too.
\( \text{new area} = k^2 \times \text{old area} \)
Area grows by \(k^2\), because BOTH dimensions are multiplied by \(k\).
\( \text{translate / reflect / rotate} \Rightarrow \text{nothing changes} \)
For the three rigid transformations, perimeter AND area stay exactly the same — the shape has only moved, flipped or turned.
\( P \rightarrow kP \qquad A \rightarrow k^2 A \)
Key idea: under enlargement, lengths scale by \(k\) but areas scale by \(k^2\) — a shape twice as long has four times the area.
\[ (x+a;\ y+b) \quad (x;\ -y) \quad (-x;\ -y) \quad (kx;\ ky) \]
You now know how to: