Similarity and congruenceIB MYP Maths Extended: Revision notes

Two shapes are congruent if they are exactly the same shape and size. One can be moved onto the other by a translation, a rotation or a reflection. Corresponding sides and angles are equal. Congruent shapes may be turned or flipped, so match the sides by length and the angles by their position, not by how the diagram looks.

Two triangles are congruent if they satisfy one of these:

• SSS: all three pairs of sides are equal.
• SAS: two pairs of sides and the included angle (between them) are equal.
• ASA or AAS: two angles and a side are equal (the side can be between the angles or not).
• RHS: both are right-angled, with equal hypotenuses and one other pair of equal sides. AAA is not a condition: it only gives the same shape. Also SSA is not enough in general. When you give a reason, write the condition and list the equal parts.

Two shapes are similar if one is an enlargement of the other: all angles are equal and all pairs of corresponding sides are in the same ratio. The ratio is the scale factor: $k=\frac{\text{length in the new shape}}{\text{length in the original}}.$ Example: a $6$ cm by $4$ cm rectangle enlarged to $15$ cm long has $k=15\div6=2.5$, so its width is $4\times2.5=10$ cm. A scale factor greater than $1$ makes a shape larger; between $0$ and $1$ it makes it smaller. For scale drawings, a scale of $1:25$ means the model is $\frac{1}{25}$ of the real length.

If the length scale factor is $k$, then

• lengths are multiplied by $k$,
• areas are multiplied by $k^2$,
• volumes are multiplied by $k^3$. Example: bottles with $k=1.5$ and small volume $500$ mL: the large volume is $500\times1.5^3=1690$ mL, and a $150$ cm$^2$ label becomes $150\times1.5^2=338$ cm$^2$. To go back from large to small, divide by $k$, $k^2$ or $k^3$.

For similar triangles, set up equal ratios of corresponding sides, for example $\frac{x}{9}=\frac{6}{4}$, then solve. Check units before you scale: change m$^2$ to cm$^2$ using $1$ m$^2=10\,000$ cm$^2$. In real-life problems (models, maps, packaging), apply the right scale factor, state the answer in context and check it is sensible. Percentage change in volume is larger than percentage change in length: a $50\%$ increase in height gives a $237.5\%$ increase in volume, not $50\%$.