At Extended level, questions may combine two of these, and enlargements may use negative or fractional scale factors.

Find the point's displacement from the centre $C$, scale that displacement by $k$, then add it back onto $C$:

For $V=(2,3)$ enlarged by $k=-2$ about $C=(1,1)$: displacement $=(1,2)$, scaled $=(-2,-4)$, so $V' = (1-2,\ 1-4) = (-1,-3)$.

1. Draw straight lines through each pair of corresponding vertices — they all meet at one point, the centre of enlargement.
2. Compare a pair of corresponding side lengths to find $|k|$; if the image lies on the opposite side of the centre from the object, $k$ is negative.

For example, if an object side of length $2$ maps to an image side of length $4$ on the opposite side of the centre, $k = -\dfrac{4}{2} = -2$.

A combined transformation is applied one step at a time, in the order stated: transform the object first, then apply the second transformation to that image, not back to the original object. Crucially, order matters — reflecting then rotating a shape does not, in general, land in the same place as rotating then reflecting it, so "describe fully the single transformation equivalent to A followed by B" is a genuinely different question from "B followed by A".

Example: Triangle $T$ has vertices $(1,1),(3,1),(1,4)$.

• Reflect in the $y$-axis, then rotate $90°$ clockwise about $(0,0)$: reflecting first sends $(x,y)\to(-x,y)$, giving vertices $(-1,1),(-3,1),(-1,4)$. Rotating that image $90°$ clockwise about the origin sends $(x,y)\to(y,-x)$, giving final vertices $(1,1),(1,3),(4,1)$.
• Rotate first, then reflect: rotating $T$ itself $90°$ clockwise gives $(1,-1),(1,-3),(4,-1)$; reflecting that in the $y$-axis gives $(-1,-1),(-1,-3),(-4,-1)$ — a completely different image from the first route, confirming the two orders are not interchangeable.

Shape $T$ has vertices $(2,2),(4,2),(2,4)$. Shape $T'$ has vertices $(-1,-1),(-5,-1),(-1,-5)$. Describe fully the single transformation mapping $T$ onto $T'$.

Solution

Lines through corresponding vertices — e.g. $(2,2)$–$(-1,-1)$ and $(4,2)$–$(-5,-1)$ — meet at $(0,0)$, so the centre is $(0,0)$. The object side from $(2,2)$ to $(4,2)$ has length $2$; the corresponding image side from $(-1,-1)$ to $(-5,-1)$ has length $4$, on the opposite side of the centre. So:

Full description: enlargement, scale factor $-2$, centre $(0,0)$.

A description that gives only "enlargement, scale factor $-2$" without the centre — or only the centre without the scale factor — will not earn full marks; every required detail must be stated.