Similar Figures
Similar figures match in shape but can differ in size: matching angles are equal and matching sides are all in the same ratio. This NSW Year 9 Core maths subtopic is Stage 5 Core content under NESA outcome MA5‑GEO‑C‑01.
You will decide whether two figures are similar by checking both conditions, test rectangles by comparing the ratios of their sides, and see that turning or flipping a figure changes nothing. You will write similarity statements such as \(\triangle ABC \sim \triangle PQR\) with the vertices in matching order, read off matching angles and sides, and explain why equal angles on their own, or sides in one ratio on their own, are not enough. You will also see that congruent figures are similar with a ratio of \(1\). It is part of Year 9 Core Congruence and Similar Figures.
Every question with a fully worked solution.
- Similar Figures - Video - Introduction to Similar Figures Watch
Theory
Similar figures have the same shape but not necessarily the same size: one is an enlargement of the other. This Year 9 maths page shows the two conditions for similarity, how to read a similarity statement such as \(ABC \sim PQR\) to match corresponding angles and sides, and how similar differs from congruent.
Two figures are similar if they have the same shape. One is an enlargement or a reduction of the other, and it may also be turned or flipped.
For two figures to be similar, both of these must be true:
- every pair of corresponding angles is equal, and
- every pair of corresponding sides is in the same ratio.
One condition on its own is not enough. All rectangles have four right angles, but they are not all similar.
The symbol \(\sim\) means “is similar to”. In a similarity statement the vertices are written in matching order, so \(ABC \sim PQR\) tells you \(A\) matches \(P\), \(B\) matches \(Q\) and \(C\) matches \(R\).
Congruent figures are similar figures that are also the same size: the ratio of their sides is \(1\). Every congruent pair is similar, but most similar pairs are not congruent.
For similar figures \(ABC \sim PQR\), every pair of matching sides has the same ratio:
| From \(ABC \sim PQR\) | Angles | Sides |
|---|---|---|
| \(A \leftrightarrow P\) | \(\angle A = \angle P\) | \(AB \leftrightarrow PQ\) |
| \(B \leftrightarrow Q\) | \(\angle B = \angle Q\) | \(BC \leftrightarrow QR\) |
| \(C \leftrightarrow R\) | \(\angle C = \angle R\) | \(CA \leftrightarrow RP\) |
How to decide if two figures are similar
- Match the corners: use the similarity statement, the angle marks, or the shortest and longest sides.
- Check the angles: every matching pair must be equal.
- Check the sides: divide each side of one figure by its matching side in the other. Every ratio must be the same.
- Conclude: similar only if both checks pass. A ratio of \(1\) means the figures are also congruent.
All the angles are right angles, so the angles match. Now compare matching sides: short with short, long with long.
| \(\dfrac{20}{5}\) | \(=\) | \(4\) |
| \(\dfrac{32}{8}\) | \(=\) | \(4\) |
Both conditions hold, so the rectangles are similar. Every side has been multiplied by \(4\).
Write the two names one above the other and read down the columns.
| \(P \ Q \ R \ S\) | \(\) | \(\) |
| \(E \ F \ G \ H\) | \(\) | \(\) |
| \(Q\) | \(\leftrightarrow\) | \(F\) |
| \(R\) | \(\leftrightarrow\) | \(G\) |
| \(S\) | \(\leftrightarrow\) | \(H\) |
So \(QR\) matches \(\boldsymbol{FG}\), and \(\angle S = \boldsymbol{\angle H}\). The picture is turned, so match by the letters, not by position.
Read the matching off the statement: \(G \leftrightarrow M\), \(H \leftrightarrow N\), \(I \leftrightarrow O\). So \(\angle H = \angle N = 61^\circ\). Then use the angle sum of \(\triangle GHI\).
| \(\angle I\) | \(=\) | \(180^\circ - 52^\circ - 61^\circ\) |
| \(\) | \(=\) | \(\mathbf{67^\circ}\) |
One counter-example is enough. Take a \(2\) cm by \(3\) cm rectangle and a \(2\) cm by \(5\) cm rectangle, and compare matching sides.
| \(\dfrac{2}{2}\) | \(=\) | \(1\) |
| \(\dfrac{5}{3}\) | \(=\) | \(1.66\ldots\) |
The ratios differ, so the sides are not in proportion. Equal angles are not enough: the sides must be in the same ratio too.
Common pitfalls
Frequently asked questions
What are similar figures?
Figures with the same shape but not necessarily the same size. Their matching angles are equal and their matching sides are in the same ratio.
How do you know if two shapes are similar?
Check that every pair of matching angles is equal, and that every side of one shape divided by its matching side in the other gives the same number. Both must be true.
What does the symbol ~ mean in maths?
It means “is similar to”. The vertices are written in matching order, so in \(\triangle ABC \sim \triangle PQR\), \(A\) matches \(P\), \(B\) matches \(Q\) and \(C\) matches \(R\).
What is the difference between similar and congruent?
Congruent figures are the same shape and the same size. Similar figures are the same shape but can be different sizes. Congruent figures are a special case of similar figures where the ratio is \(1\).
Are all squares similar?
Yes. Every square has four right angles and four equal sides, so any two squares have equal angles and sides in the same ratio.