C7.1 Transformations

Learning objectives

C7.1.1Reflection of a shape in a vertical or horizontal line.• Reflection of a shape in a vertical or horizontal line.C7.1.2Rotation of a shape about the origin, vertices or midpoints of edges of• Rotation of a shape about the origin, vertices or midpoints of edges of the shape, through multiples of 90°.C7.1.3Enlargement of a shape from a centre by a scale factor.• Enlargement of a shape from a centre by a scale factor.C7.1.4Translation 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 will not involve combinations of transformations. A ruler must be used for all straight edges. Positive and fractional scale factors only.

A reflection places every image point directly across the mirror line at the same perpendicular distance. The mirror line is the perpendicular bisector of each point-to-image segment.

Mirror line	Point rule	Coordinate that stays fixed
$x=a$	$(x,y)\mapsto(2a-x,y)$	$y$
$y=b$	$(x,y)\mapsto(x,2b-y)$	$x$

Reflect each vertex separately, checking equal grid distance on opposite sides of the line, then join the image vertices with a ruler in the original order. A point on the mirror line does not move.

Reflect $P(-6,-2)$ in $y=-3$ . The $x$ -coordinate stays $-6$ , and $y'=2(-3)-(-2)=-4$ , so $P'=(-6,-4)$ . Both points are one unit from the mirror line.

A full description names both reflection and its exact line, such as $x=-2$ or $y=4$ . Do not swap coordinates unless a different diagonal mirror line is explicitly in scope; this Core objective uses horizontal or vertical lines only.

A rotation turns every point through the same angle and direction around one fixed centre. Distance from the centre is preserved.

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 a centre other than the origin, subtract the centre coordinates from each point, apply the relevant origin rule, then add the centre back. Rotate all vertices and join them in order with a ruler.

Rotate $P(7,3)$ by $90^\circ$ anticlockwise about $(5,1)$ . Relative to the centre, $P$ is $(2,2)$ ; this becomes $(-2,2)$ , so $P'=(5-2,1+2)=(3,3)$ .

A full description needs rotation, centre, angle and clockwise/anticlockwise direction; direction is unnecessary for $180^\circ$ . The permitted angles are multiples of $90^\circ$ , and the shape must not change size.

An enlargement sends each point along the ray from a fixed centre, multiplying its distance from that centre by the scale factor $k$ .

P'=C+k(P-C)

Draw a ray from the centre through each original vertex. If $k>1$ , place the image farther along the ray; if $0<k<1$ , place it between the centre and the original point. Repeat for every vertex and join with a ruler. To find the centre from two shapes, extend corresponding point-to-image lines until they meet.

With centre $C=(4,0)$ , point $P=(-6,-4)$ and $k=\frac12$ , the centre-to-point vector is $(-10,-4)$ . Half is $(-5,-2)$ , so $P'=C+(-5,-2)=(-1,-2)$ .

A full description names enlargement, centre and scale factor. This Core scope uses positive scale factors, including fractions: orientation stays the same, lengths multiply by $k$ , and area multiplies by $k^2$ .

A translation moves every point by the same vector, so the image keeps the original size, shape and orientation.

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

The top component is horizontal: positive means right and negative means left. The bottom component is vertical: positive means up and negative means down. Add the components to every vertex, then join corresponding image vertices with a ruler.

Translating $P(-4,3)$ by $\begin{pmatrix}-3\\-4\end{pmatrix}$ gives $P'=(-4-3,3-4)=(-7,-1)$ . To describe a translation from $P$ to $P'$ , subtract original coordinates from image coordinates in the same order.

A translation vector describes movement, not the final coordinates. All points must move by exactly the same vector. Its movement length is $\sqrt{a^2+b^2}$ , but that length alone does not describe direction. Questions here use one transformation, not a combination.

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—Cambridge IGCSE Mathematics (0580)Read

C7.1 Transformations

Learning objectives

Reflect a shape in a horizontal or vertical line

Rotate a shape through multiples of 90°

Enlarge a shape from a centre

Translate a shape by a vector

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