5.2 Transformation geometry

Learning objectives

5.2A Specifying rotationsUnderstand that a rotation is specified by a centre and an angle.5.2B Rotate a shape about a point through a given anglerotate a shape about a point through a given angle 5.2C Positive and negative rotationsrecognise that an anti-clockwise rotation is a positive angle of rotation and a clockwise rotation is a negative angle of rotation 5.2D Reflection in a mirror lineunderstand that reflections are specified by a mirror line Such as x = 1, y = 2, y = x, y – x = 0 5.2E Constructing mirror lines and reflectionsConstruct a mirror line from an object and its image, and reflect a shape in a given mirror line.5.2F Translation by distance and directionunderstand that translations are specified by a distance and direction 5.2G Translate a shapetranslate a shape 5.2H Column vectors in translationsunderstand and use column vectors in translations 5.2I Rigid transformations and congruenceUnderstand that rotations, reflections and translations preserve lengths and angles, so the image remains congruent to the original.5.2J Specifying enlargementsUnderstand that an enlargement is specified by a centre and a positive scale factor, including fractional scale factors.5.2K Angle and length effects of enlargementUnderstand that enlargements preserve angles but do not generally preserve lengths.5.2L Enlarge a shape given the scale factor With or withoutenlarge a shape given the scale factor With or without a centre given 5.2M Complete descriptions of transformationsIdentify transformations and give complete descriptions of them.

A rotation turns every point through the same angle about one fixed centre. A complete specification states the centre, angle and direction.

Required part	Meaning
centre	fixed point of the turn
angle	amount of turn
direction	clockwise or anticlockwise

Every point and its image are the same distance from the centre, and the angle between their centre-lines is the rotation angle.

Saying only 'rotation' or giving an angle without a centre is incomplete.

Rotate each vertex about the stated centre, keeping its distance from the centre unchanged, then join the image vertices in the original order.

Step	Action
1	mark the centre
2	trace centre-to-vertex displacement
3	turn that displacement through the given angle
4	plot the image vertex and repeat

For a $180^\circ$ turn about $(a,b)$ , $(x,y)$ maps to $(2a-x,2b-y)$ .

Do not rotate about the origin unless the stated centre is the origin.

By convention, anticlockwise rotations have positive angles and clockwise rotations have negative angles.

Description	Equivalent description
$90^\circ$ clockwise	$-90^\circ$ or $270^\circ$ anticlockwise
$90^\circ$ anticlockwise	$+90^\circ$ or $270^\circ$ clockwise
$180^\circ$	same result in either direction

Keep the centre fixed while deciding direction; the sign describes the turn, not a coordinate sign.

Positive does not mean clockwise, and a full $360^\circ$ change gives the original position.

A reflection maps every point across a mirror line. The mirror line is the perpendicular bisector of the segment joining a point to its image.

Mirror line	Coordinate effect
$x=a$	horizontal distance to $x=a$ changes side
$y=b$	vertical distance to $y=b$ changes side
$y=x$	$(x,y)\mapsto(y,x)$
$y-x=0$	same line as $y=x$

A point on the mirror line stays fixed; paired points lie at equal perpendicular distances on opposite sides.

The mirror line is not usually the line joining a point to its image; it crosses that segment at right angles halfway along.

To reflect a shape, send each vertex along a perpendicular to the mirror line by the same distance to the opposite side. To recover the mirror line, construct perpendicular bisectors of point-image pairs.

Given	Construction
mirror line	measure perpendicular distance for every vertex, copy it across
object and image	join matching vertices, mark midpoints, draw their common perpendicular bisector

Join the reflected vertices in corresponding order and verify that lengths and angles match the original.

Measuring horizontal or vertical distance works only for vertical or horizontal mirror lines; oblique lines require perpendicular distance.

A translation slides every point by the same directed displacement. Its distance and direction are identical for all corresponding point pairs.

Feature	Translation effect
movement	same distance and direction for every point
orientation	unchanged
fixed centre or line	none required

Compare any vertex with its image: the horizontal and vertical changes must match those for every other vertex.

A translation does not turn, flip or resize the shape.

Choose each vertex, apply the same horizontal and vertical displacement, plot its image, then reconnect the vertices in the same order.

Step	Action
1	identify a matching start vertex
2	count horizontal movement
3	count vertical movement
4	repeat exactly for every vertex

Corresponding sides remain parallel and equal, and the image has the same orientation and size.

Do not repeatedly move from the previous image vertex; apply the displacement independently to each original vertex.

The translation vector $\begin{pmatrix}a\\b\end{pmatrix}$ moves each point $a$ units horizontally and $b$ units vertically: $(x,y)\mapsto(x+a,y+b)$ .

Entry	Positive	Negative
top, $a$	right	left
bottom, $b$	up	down

The vector $\begin{pmatrix}-4\\3\end{pmatrix}$ means 4 left and 3 up.

A translation vector uses a column, not coordinate notation; its entries describe a change rather than a location.

Rotations, reflections and translations are rigid transformations: they preserve all lengths and angles, so the image is congruent to the original.

Transformation	Lengths	Angles	Orientation
rotation	preserved	preserved	preserved
translation	preserved	preserved	preserved
reflection	preserved	preserved	reversed

Corresponding side lengths, angle sizes, perimeter and area are unchanged under a rigid transformation.

Congruent does not mean identical position or orientation; an enlargement with scale factor other than 1 is not rigid.

An enlargement is specified by a centre and a positive scale factor $k$ . Each image point lies on the ray from the centre through the original point, at $k$ times the original distance.

Scale factor	Effect
$k>1$	image farther from centre and larger
$k=1$	unchanged
$0<k<1$	image between centre and original, smaller

Lines joining corresponding vertices pass through the centre of enlargement and their distance ratios equal $k$ .

This syllabus specifies positive scale factors only; do not introduce negative enlargements into this objective.

An enlargement preserves corresponding angles and multiplies every corresponding length by the same scale factor $k$ .

Measure	Factor
angle	unchanged
length and perimeter	$k$
area	$k^2$

The image is similar to the original. It is congruent only when $k=1$ .

Preserved angles do not imply preserved lengths; for $k=3$ , every length triples.

For each vertex, draw or imagine a ray from the centre through that vertex and place the image at $k$ times the centre-to-vertex distance.

Step	Action
1	locate the centre
2	form centre-to-vertex rays
3	multiply each displacement by $k$
4	join image vertices in order

With centre $(a,b)$ , $(x,y)$ maps to $(a+k(x-a),\ b+k(y-b))$ .

Multiplying coordinates directly by $k$ works only when the centre is the origin.

Identify the single transformation by comparing size, orientation and point movement, then state every parameter required for that type.

Type	Complete description needs
translation	column vector
reflection	mirror-line equation
rotation	centre, angle and direction
enlargement	centre and positive scale factor

Equal size suggests a rigid transformation; reversed orientation suggests reflection; changed size suggests enlargement. Confirm with corresponding vertices before naming it.

Do not list several transformations when asked for a single one, and do not confuse a centre coordinate $(a,b)$ with a translation vector $\begin{pmatrix}a\\b\end{pmatrix}$ .

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—Pearson Edexcel International GCSE Mathematics (Specification A) (4MA1)Read

5.2 Transformation geometry

Learning objectives

Specify a rotation completely

Rotate a shape about a given point

Use signs for rotation direction

Specify a reflection by its mirror line

Construct mirror lines and reflected shapes

Describe a translation by distance and direction

Translate a shape

Use column vectors for translations

Recognise rigid transformations and congruence

Specify an enlargement completely

Track angle and length effects of enlargement

Enlarge a shape from a centre and scale factor

Give complete transformation descriptions

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