For similar figures with linear scale factor $k$: $$\text{Length ratio} = k \qquad \text{Area ratio} = k^2 \qquad \text{Volume ratio} = k^3$$ So $k = \dfrac{\text{new length}}{\text{old length}}$, and $k = \sqrt{\dfrac{A_2}{A_1}} = \sqrt[3]{\dfrac{V_2}{V_1}}$.

Enlargement (centre $O$, scale factor $k$)

$$\text{Image distance from centre} = k \times \text{object distance from centre}$$

• $k > 1$: image larger. $0 < k < 1$: image smaller. $k < 0$: image on the opposite side of the centre, inverted.
• To find the centre, join corresponding points; the lines meet at the centre.
• Area scale factor of an enlargement $= k^2$.

Congruence tests

SSS, SAS, ASA (or AAS), RHS.