Transformations Notes

Q: How do you describe a rotation?

State the centre, the angle and the direction. A rotation turns a shape about a fixed centre through an angle, clockwise or anticlockwise. All three must be stated (a 180° turn needs no direction).

Q: How do you spot which transformation has happened?

Check size first, then whether it is flipped, then whether it is turned. Compare object and image. Different size means enlargement (compare sides to find k; if upside down, k is negative). Same size but mirror image means reflection (find the line midway between corresponding points).

Edexcel IGCSE Maths: Revision notes

Key facts

Translation: a column vector $\begin{pmatrix}x\\y\end{pmatrix}$ ; congruent, same orientation.
Reflection: the equation of the mirror line; congruent, but a mirror image.
Rotation: centre, angle and direction (not needed for $180°$ ); congruent, same orientation.
Enlargement: centre and scale factor $k$ ; sides $\times k$ , area $\times k^2$ ; a negative $k$ flips through the centre.
All four preserve angles; only enlargement changes lengths and area.

Translation

A translation slides every point the same distance in the same direction, described by a column vector.

A translation slides a shape without turning or resizing it. The column vector $\begin{pmatrix}x\\y\end{pmatrix}$ gives the horizontal shift (positive is right) over the vertical shift (positive is up). The image is congruent to the object.

Triangle ABC translated by (2, −3) to A′B′C′

Worked example

Translate the triangle with vertices $A(1,1)$ , $B(3,1)$ , $C(1,4)$ by $\begin{pmatrix}2\\-3\end{pmatrix}$ .

1
Add the vector to every vertex.
2
$A\to(3,-2)$ , $B\to(5,-2)$ , $C\to(3,1)$ .

What does the vector $\begin{pmatrix}-4\\5\end{pmatrix}$ do to a point?

Reflection

A reflection flips a shape in a mirror line; state the equation of the line.

A reflection flips a shape. Every point and its image are the same perpendicular distance from the mirror line. Useful rules:

In $y=x$ : $(a,b)\to(b,a)$
In $y=-x$ : $(a,b)\to(-b,-a)$
In the $x$ -axis: $(a,b)\to(a,-b)$
In the $y$ -axis: $(a,b)\to(-a,b)$

The image is congruent but has opposite orientation: a mirror image.

Triangle reflected in the mirror line y = x

Worked example

Reflect the point $P(3,5)$ in the line $x=1$ .

1
The distance from $P$ to the line is $3-1=2$ .
2
The image is 2 units the other side: $x=1-2=-1$ . The $y$ -coordinate is unchanged.

What is the image of $(2,7)$ when reflected in the line $y=x$ ?

Rotation

State the centre, the angle and the direction.

A rotation turns a shape about a fixed centre through an angle, clockwise or anticlockwise. All three must be stated (a $180°$ turn needs no direction).

About the origin:

$90°$ clockwise: $(a,b)\to(b,-a)$
$90°$ anticlockwise: $(a,b)\to(-b,a)$
$180°$ : $(a,b)\to(-a,-b)$

For another centre, subtract the centre, rotate, then add it back. The image is congruent with the same orientation.

Rotating (5, 3) by 90° clockwise about C(2, 1) gives (4, −2)

Worked example

Rotate $(5,3)$ by $90°$ clockwise about $(2,1)$ .

1
Subtract the centre: $(3,2)$ .
2
Apply $(a,b)\to(b,-a)$ : $(2,-3)$ .
3
Add the centre back: $(4,-2)$ .

What is the image of $(2,5)$ after a $180°$ rotation about the origin?

Enlargement

Multiply the vector from the centre by k, then add it back to the centre.

An enlargement resizes from a centre by scale factor $k$ ; state both.

$k>1$ : bigger, same side of the centre
$0<k<1$ : smaller, same side
$k<0$ : resized and flipped through the centre, upside down on the opposite side

Unlike the other three, an enlargement with $k\neq\pm1$ is not congruent but similar: sides $\times k$ , area $\times k^2$ .

Enlargement, scale factor 2, centre O(1, 1)

Worked example

Enlarge the point $(3,1)$ by scale factor $-2$ from centre $(1,1)$ .

1
Vector from the centre: $(3-1,1-1)=(2,0)$ .
2
Multiply by $-2$ : $(-4,0)$ .
3
Add to the centre: $(1-4,1)$ .

A shape is enlarged with scale factor $\frac{1}{2}$ . What happens to it?

Spotting the transformation

Check size first, then whether it is flipped, then whether it is turned.

Compare object and image. Different size means enlargement (compare sides to find $k$ ; if upside down, $k$ is negative). Same size but mirror image means reflection (find the line midway between corresponding points). Same size, turned, same orientation means rotation (find the fixed centre). Otherwise it is a translation (find the vector between corresponding vertices).

Combined transformations: describe each one fully and in order.

1
Different size?
enlargement: find the centre and k
2
Flipped (mirror image)?
reflection: find the mirror line
3
Turned?
rotation: find centre, angle and direction
4
Just moved?
translation: find the column vector

Spotting the transformation

An image is the same size as the object but is a mirror image of it. Which transformation is it?

Try an exam question

(a) Reflect the point $P(3,5)$ in the line $x=1$ . (b) Enlarge the point $(3,1)$ by scale factor $-2$ with centre $(1,1)$ .

[4 marks]

That's the notes covered.

Carry on to the next subtopic.