Similarity

Similar figures have:

• Equal corresponding angles.
• Corresponding sides in proportion — there's a single scale factor $k$ that converts one figure's sides into the other's.

Notation: $\triangle ABC \sim \triangle DEF$ means triangle $ABC$ is similar to triangle $DEF$, with $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$.

Conditions for similarity (triangles).

• AA (angle-angle): two pairs of equal angles. The third pair is automatic ($180\degree$ sum).
• SSS proportional: all three pairs of sides in the same ratio.
• SAS proportional: two pairs of sides in the same ratio AND the included angle equal.

Worked. Two triangles share two equal angles ($60\degree$ and $80\degree$). Are they similar?

• AA satisfied → similar.

Worked. Triangle $A$ has sides $4, 6, 8$. Triangle $B$ has sides $6, 9, 12$. Are they similar?

• Ratios: $\dfrac{6}{4} = 1.5$, $\dfrac{9}{6} = 1.5$, $\dfrac{12}{8} = 1.5$. All equal.
• Yes, similar (SSS proportional). Scale factor $k = 1.5$ from $A$ to $B$.

Method.

1. Identify which figures are similar (often given).
2. Match corresponding vertices.
3. Write equal ratios of corresponding sides.
4. Cross-multiply to find the unknown.

Worked. Triangles $ABC$ and $DEF$ are similar with $A \leftrightarrow D$, etc. $AB = 6$, $DE = 9$, $BC = 8$. Find $EF$.

• $\dfrac{AB}{DE} = \dfrac{BC}{EF}$.
• $\dfrac{6}{9} = \dfrac{8}{EF}$.
• $EF = \dfrac{8 \times 9}{6} = 12$.

Tip. Sketch the two triangles with corresponding sides aligned the same way. Then it's clear which side maps to which.

If two figures are similar with linear scale factor $k$:

• Area scale factor $= k^{2}$.
• Volume scale factor $= k^{3}$.

Worked. Two similar triangles. The smaller has area $20\,\text{cm}^{2}$. Lengths in the larger are $3$ times those in the smaller. Find the area of the larger.

• $k = 3$. Area scale: $k^{2} = 9$.
• Larger area: $20 \times 9 = 180\,\text{cm}^{2}$.

Worked. Two similar cylinders. Volume of smaller is $50\,\text{cm}^{3}$. The larger has lengths $2$ times bigger. Volume of larger?

• $k = 2$. Volume scale: $k^{3} = 8$.
• Larger volume: $50 \times 8 = 400\,\text{cm}^{3}$.

Reverse direction. "Two similar triangles have areas $25$ and $100$. Find the linear scale factor."

• Area ratio: $4$. Linear ratio: $\sqrt{4} = 2$.

Reverse for volume. Volume ratio $27$ → linear ratio $\sqrt[3]{27} = 3$.

When parallel lines cut across two converging transversals (like rays from a single point), they create similar triangles by AA.

Worked example. A small triangle is nested inside a larger one, sharing the same apex, with a base parallel to the larger triangle's base. The two triangles are similar.

If the smaller triangle has base $4$ and height $3$, and the larger has height $9$:

• Linear scale $k = 9/3 = 3$.
• Larger base = $3 \times 4 = 12$.

Common scenario. Cambridge gives you a triangle with a line drawn parallel to one side, dividing the triangle into a smaller triangle plus a trapezium. The smaller triangle is similar to the whole.