Congruent and similar figures: correspondence, conditions and scale

5 min read·Updated 5 September 2026

Questions on congruence and similarity almost never ask you to prove anything in Paper 2 — there is no room for a proof in ninety seconds. They ask you to use a relationship that has already been handed to you, or to notice that one is there. Typically you are given two triangles (or two solids) and told they are similar, or given enough angle information that they must be, and asked for a missing length, a ratio of areas, or which of four statements about them is true.

The whole topic runs on one sentence: similar figures have equal corresponding angles and a constant ratio between corresponding sides. Congruence is the special case where that ratio is one. Everything that goes wrong goes wrong at the word corresponding.

Misconception 1: reading the correspondence off the picture

A similarity statement is not a list of two shapes. It is a pairing. Writing that triangle ABC is similar to triangle PQR says A pairs with P, B with Q, C with R — and therefore that AB pairs with PQ, not with QR. Swap two letters and every ratio you write afterwards is a ratio of sides that have nothing to do with one another.

This is the single most dangerous error in the topic because it does not feel like an error. You still divide one side by another, you still get a clean number, and the number still looks like a scale factor. Nothing in the arithmetic complains. The only defence is to write the pairing down before you write the ratio, taking it from the letters in the statement, never from which sides happen to look similar in the diagram.

Misconception 2: similarity by family resemblance

Two triangles that share one angle are not similar. Two quadrilaterals with the same angles are not similar either — a square and a non-square rectangle have four right angles each and are nothing like each other. A drawing that looks like a scaled copy is not evidence, because diagrams in a multiple-choice paper are not drawn to scale and are frequently drawn to mislead.

What actually licenses similarity is a stated condition: two pairs of equal angles, or three pairs of sides in a constant ratio, or two pairs of sides in a constant ratio with the angle between them equal. If none of those is established, the figures are just two shapes in the same picture.

Misconception 3: carrying a length ratio into an area

If corresponding lengths are in the ratio $k$, corresponding areas are in the ratio $k^2$ and corresponding volumes in the ratio $k^3$. Students who know this rule perfectly well still forget to apply it, because the question asks for an area and the ratio they have just computed is sitting right there, ready to multiply. The rule is not arbitrary: an area is a product of two lengths, so scaling every length by $k$ scales the product by $k \times k$.

A worked example

Triangle ABC is similar to triangle PQR, with $AB = 6$ cm and $PQ = 9$ cm. The area of triangle ABC is $20$ cm². Find the area of triangle PQR.

The correspondence in the statement pairs AB with PQ, so the length scale factor from ABC to PQR is

The wrong turn here is to stop and write $20 \times \tfrac{3}{2} = 30$ cm². That answer is what you would get if area scaled like length — and it is almost always one of the four options, because it is the answer of a student who did everything right except the last step. Areas scale by the square:

A second wrong turn is subtler. If the statement had instead paired ABC with QRP, then AB would correspond to QR, not PQ, and $9$ would be the wrong number to divide by entirely. Check the letters.

Check yourself

Before any ratio, write the three pairs as a column: A–P, B–Q, C–R. Then ask "lengths, areas, or volumes?" and attach the matching power of $k$. If your final answer changed by the same factor as your lengths did, and the question asked for an area, you have not finished.

The drill below builds fresh figures every time, so you can practise reading correspondence rather than remembering it.

Original study notes by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.