E7.1 Transformations

Learning objectives

E7.1.1Reflection of a shape in a straight line.• Reflection of a shape in a straight line.E7.1.2Rotation of a shape about a centre through multiples of 90°.• Rotation of a shape about a centre through multiples of 90°.E7.1.3Enlargement of a shape from a centre by a scale factor.• Enlargement of a shape from a centre by a scale factor.E7.1.4Translation of a shape by a vector x y J L KK N P OO• Translation of a shape by a vector x y J L KK N P OO. Questions may involve combinations of transformations. A ruler must be used for all straight edges. Positive, fractional and negative scale factors may be used.

A reflection places every image point on the opposite side of the mirror line at the same perpendicular distance. The mirror line is therefore the perpendicular bisector of every point-to-image segment.

Mirror line	Point rule
$x=a$	$(x,y)\mapsto(2a-x,y)$
$y=b$	$(x,y)\mapsto(x,2b-y)$
$y=x$	$(x,y)\mapsto(y,x)$
$y=-x$	$(x,y)\mapsto(-y,-x)$
$y=x+c$	$(x,y)\mapsto(y-c,x+c)$
$y=-x+c$	$(x,y)\mapsto(c-y,c-x)$

For any other straight mirror line, draw the perpendicular from each vertex to the line, measure its distance, and mark the image the same distance beyond the line. A vertex on the mirror line stays fixed. Join image vertices in the same order with a ruler.

Reflect $P(4,-1)$ in $y=x+2$ . Using $(x,y)\mapsto(y-2,x+2)$ gives $P'=(-3,6)$ . The midpoint $\left(\frac12,\frac52\right)$ lies on $y=x+2$ , and $PP'$ is perpendicular to that line, confirming the result.

A complete description states reflection and the exact mirror line. Swapping coordinates works only for $y=x$ ; an offset diagonal line requires its offset rule or the perpendicular-distance construction.

A rotation turns every point through the same angle and direction around one fixed centre. Each point keeps its distance from the centre, so lengths, angles and area are unchanged.

Rotation about the origin	Coordinate rule
$90^\circ$ anticlockwise	$(x,y)\mapsto(-y,x)$
$90^\circ$ clockwise	$(x,y)\mapsto(y,-x)$
$180^\circ$	$(x,y)\mapsto(-x,-y)$

For centre $C=(h,k)$ , first subtract $C$ from a vertex, rotate that centre-to-point vector using the table, then add $C$ back. Repeat for every vertex and join the images in order with a ruler.

Rotate $P(7,3)$ by $90^\circ$ anticlockwise about $C=(5,1)$ . The relative vector is $(2,2)$ ; it rotates to $(-2,2)$ , so $P'=C+(-2,2)=(3,3)$ . Both $P$ and $P'$ are $\sqrt8$ units from $C$ .

A complete description gives rotation, centre, angle and clockwise or anticlockwise direction; direction need not be stated for $180^\circ$ . Do not use an origin rule on raw coordinates when the stated centre is elsewhere.

An enlargement with centre $C$ multiplies every centre-to-point vector by the same scale factor $k$ . This single rule controls the image's size, side of the centre and exact position.

P'=C+k(P-C)

Scale factor	Position of image point	Length multiplier
$k>1$	same ray, beyond the original	$k$
$0<k<1$	same ray, between centre and original	$k$
$k<0$	opposite ray from the centre	$\|k\|$

Let $C=(1,0)$ , $P=(2,1)$ and $k=-2$ . Since $P-C=(1,1)$ , multiplying gives $(-2,-2)$ and $P'=C+(-2,-2)=(-1,-2)$ . The image is opposite $P$ across $C$ and twice as far away.

A complete description gives enlargement, centre and signed scale factor. Lengths multiply by $|k|$ and areas by $k^2$ ; multiplying the raw coordinates by $k$ is valid only when the centre is the origin. Corresponding point lines meet at the centre.

A translation adds the same directed movement to every point. In a combination, perform each named transformation on the current image in the stated order; the output of one step is the input to the next.

\begin{pmatrix}x\y\end{pmatrix}\mapsto\begin{pmatrix}x+a\y+b\end{pmatrix}\quad\text{for translation vector }\begin{pmatrix}a\b\end{pmatrix}

Read the top component as horizontal movement and the bottom as vertical movement. Label an intermediate image after every transformation, apply the next rule to all of its vertices, and join straight edges with a ruler. Keeping intermediate coordinates prevents the original shape being transformed twice.

Start with $P=(2,1)$ . Reflect it in $y=x$ to obtain $P_1=(1,2)$ , then translate by $\begin{pmatrix}-3\\4\end{pmatrix}$ to obtain $P_2=(-2,6)$ . Reversing the order gives a different result, so the written sequence matters.

A vector records movement, not destination coordinates. Translation alone preserves size, shape and orientation. A combination must be described step by step: there is generally no single equivalent named transformation, and changing the order can change the final image.

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—E7.2 Vectors in two dimensionsRead

E7.1 Transformations

Learning objectives

Reflect a shape in any straight line

Rotate a shape about a stated centre

Enlarge with positive, fractional or negative scale factor

Translate by a vector and combine transformations

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