Enlargement by rational scale factorsIB MYP Maths Extended: Revision notes

An enlargement changes the size of a shape but keeps its shape. It is described by two things: the centre of enlargement (a fixed point) and the scale factor $k$. Every length is multiplied by $|k|$, but angles stay the same, so the image is similar to the object. A scale factor greater than 1 makes the image bigger. A scale factor between 0 and 1, such as $\frac12$ or $\frac23$, makes the image smaller; it is still called an enlargement. A scale factor of 1 leaves the shape unchanged.

Measure from the centre. Each point of the image lies on the straight line through the centre and the object point, at $k$ times the distance from the centre. With coordinates, use the vector from the centre $C$ to each point $P$: $\text{image}=C+k\,(P-C).$ If the centre is the origin, just multiply each coordinate by $k$. Worked example. Enlarge $P(3,2)$ with centre $(1,0)$ and scale factor $\frac12$. The vector from the centre is $(2,2)$; half of it is $(1,1)$; add it to the centre: $(1,0)+(1,1)=(2,1)$.

When $k$ is negative, the image is on the opposite side of the centre from the object, and the shape is turned upside down (it looks rotated by $180^\circ$). The size still changes by $|k|$. For centre $O$ and $k=-2$, the point $(3,1)$ goes to $(-6,-2)$: twice as far from $O$, on the other side. With centre $(1,1)$ and $k=-2$, the point $(3,1)$ has vector $(2,0)$ from the centre, so the image is $(1,1)-2(2,0)=(-3,1)$.

To describe an enlargement you must give the word enlargement, the scale factor and the centre.

• Scale factor: $\dfrac{\text{image length}}{\text{object length}}$. Make it negative if the image is on the opposite side of the centre and inverted.
• Centre: draw a straight line through each pair of corresponding points. The lines meet at the centre. Example: $T$ with vertices $(2,1)$, $(3,1)$, $(3,3)$ maps to $(-1,1)$, $(-3,1)$, $(-3,-3)$. Lengths double and the image is inverted, so $k=-2$. The lines $y=1$ and $y=x$ meet at the centre $(1,1)$.

If the scale factor is $k$:

• lengths are multiplied by $|k|$
• areas are multiplied by $k^2$
• volumes are multiplied by $|k|^3$. The area and volume factors are always positive, even if $k$ is negative. Worked example. A model is an enlargement of a statue with $k=\frac1{20}$. The statue's surface area is 48 m$^2$ and its volume is 21 m$^3$. Model area $=48\times\frac1{400}=0.12$ m$^2$. Model volume $=21\times\frac1{8000}=0.002625$ m$^3$. To go from the image back to the object, divide by the same factor.