Similar Figures
Similar figures have the same shape, though their sizes can differ: matching angles are equal and matching sides are in the same ratio. This NSW Year 9 Advanced maths subtopic writes similarity statements with the symbol \(\sim\) (NESA outcome MA5‑GEO‑C‑01).
You will match corresponding angles and sides, decide whether rectangles and triangles are similar, and find a scale factor. You will decide whether a square and a rhombus with sides in the same ratio are similar, find both possible sides of a similar rectangle, test whether a frame is similar to its photo, and show why A4 paper keeps its shape when it is halved. It is part of Year 9 Advanced 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 or reduction of the other. Matching angles are equal and matching sides are all multiplied by the same scale factor. This Year 9 Advanced maths page covers similarity statements, testing for similarity and finding unknowns.
Two figures are similar if one is an enlargement or reduction of the other, possibly turned or flipped as well. They have the same shape.
In similar figures, corresponding angles are equal and corresponding sides are in the same ratio. That common ratio is the scale factor \(k\). If \(k > 1\) it is an enlargement; if \(0 < k < 1\) it is a reduction.
The symbol \(\sim\) means "is similar to". As with congruence, a similarity statement lists the vertices in matching order: \(\triangle ABC \sim \triangle PQR\) means \(A\) matches \(P\), \(B\) matches \(Q\) and \(C\) matches \(R\).
Congruent figures are similar with \(k = 1\). Every pair of squares, of circles and of equilateral triangles is similar. Rectangles, rhombuses and other shapes are similar only when both conditions hold.
Scale factor from figure 1 to figure 2:
If \(\triangle ABC \sim \triangle PQR\):
How to test whether two figures are similar
- Pair up the sides: shortest with shortest, longest with longest (or use a given statement or diagram).
- Work out each ratio new \(\div\) old. Fractions in simplest form or exact decimals avoid rounding errors.
- Compare: all ratios equal (and, for shapes other than triangles, all angles equal) means similar.
- To show they are not similar, one unequal ratio or one unequal angle is enough. A single counterexample disproves a general claim.
How to find an unknown length
- Match the vertices from the statement or the equal angles.
- Find \(k\) from a pair of matching sides that are both known.
- Multiply an original length by \(k\), or divide a new length by \(k\) to go back.
- Not told which sides match? Try each possibility; there may be two answers.
- Algebra: set two ratios equal, cross-multiply and solve. Reject answers that make a length zero or negative.
(i) Find the third angle of each triangle.
(ii) Write a similarity statement with the vertices in matching order.
| \(\angle R\) | \(=\) | \(180^\circ - 38^\circ - 105^\circ = 37^\circ\) |
| \(\angle X\) | \(=\) | \(180^\circ - 38^\circ - 37^\circ = 105^\circ\) |
(ii) Match equal angles: \(P \leftrightarrow Y\) (\(38^\circ\)), \(Q \leftrightarrow X\) (\(105^\circ\)), \(R \leftrightarrow Z\) (\(37^\circ\)). Write them in the same positions:
(i) Show the triangles are similar and find the scale factor.
(ii) Write the similarity statement.
(i) Pair the sides in order of size: shortest with shortest, and so on.
| \(\dfrac{LM}{AB}\) | \(=\) | \(\dfrac{15}{6} = 2.5\) |
| \(\dfrac{KL}{BC}\) | \(=\) | \(\dfrac{22.5}{9} = 2.5\) |
| \(\dfrac{MK}{CA}\) | \(=\) | \(\dfrac{30}{12} = 2.5\) |
All three ratios are equal, so the triangles are similar with scale factor \(\mathbf{2.5}\).
(ii) A vertex is where two sides meet. \(B\) joins \(AB\) and \(BC\); their partners \(LM\) and \(KL\) meet at \(L\). So \(B \leftrightarrow L\). \(A\) joins \(AB\) and \(CA\); \(LM\) and \(MK\) meet at \(M\). So \(A \leftrightarrow M\), and \(C \leftrightarrow K\).
(i) Find the scale factor from \(ABCD\) to \(EFGH\).
(ii) Find \(x\).
(iii) Find \(y\) as a mixed number.
(i) \(EF\) matches \(AB\) (1st and 2nd letters). Divide new by old.
| \(k\) | \(=\) | \(\dfrac{EF}{AB} = \dfrac{8}{12} = \mathbf{\dfrac{2}{3}}\) |
(ii) \(FG\) matches \(BC\).
| \(x - 1\) | \(=\) | \(9 \times \dfrac{2}{3} = 6\) |
| \(x\) | \(=\) | \(\mathbf{7}\) |
(iii) \(GH\) matches \(CD\).
| \(y\) | \(=\) | \(10 \times \dfrac{2}{3} = \dfrac{20}{3} = \mathbf{6\dfrac{2}{3}}\) |
A scale factor less than \(1\) is a reduction: every side of \(EFGH\) is shorter.
The leftover is \(12\) cm by \((20 - x)\) cm. Its long side is \(12\), which matches the original's long side \(20\). So its short side \(20 - x\) matches \(12\).
| \(\dfrac{20 - x}{12}\) | \(=\) | \(\dfrac{12}{20}\) |
| \(20 - x\) | \(=\) | \(\dfrac{144}{20} = 7.2\) |
| \(x\) | \(=\) | \(\mathbf{12.8}\) |
Check: \(12 \div 7.2 = 1.6\overline{6}\) and \(20 \div 12 = 1.6\overline{6}\) \(\checkmark\)
Common pitfalls
Frequently asked questions
What does similar mean in maths?
Similar figures have the same shape but not necessarily the same size. Their matching angles are equal and their matching sides are all in the same ratio.
How do you find the scale factor of similar figures?
Divide a length in the new figure by the matching length in the original figure. A scale factor greater than \(1\) is an enlargement and less than \(1\) is a reduction.
How do you know if two shapes are similar?
Check that every pair of matching angles is equal and every pair of matching sides has the same ratio. For triangles, equal angles or equal side ratios on their own are enough.
Are all rectangles similar?
No. All rectangles have right angles, but their sides are not always in the same ratio. A \(2\) by \(3\) rectangle and a \(2\) by \(5\) rectangle are not similar. All squares are similar.
What is the symbol for similar?
The symbol is \(\sim\), as in \(\triangle ABC \sim \triangle PQR\). The vertices are written in matching order.
Are congruent shapes similar?
Yes. Congruent shapes are similar with a scale factor of \(1\). But similar shapes are only congruent when the scale factor is \(1\).