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SingaporeMathsSyllabus dot point

When are two figures congruent or similar, and how do we use similarity to find missing lengths?

Identify congruent and similar figures, and use the ratio of corresponding sides of similar figures to find unknown lengths

A focused answer to the N(A)-Level Mathematics outcome on congruence and similarity. The meaning of congruent and similar figures, equal angles and proportional sides, and finding missing lengths by scale factor.

Generated by Claude Opus 4.88 min answer

Reviewed by: AI editorial process; not yet individually human-reviewed

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  1. What this dot point is asking
  2. The answer
  3. Examples in context
  4. Try this

What this dot point is asking

SEAB wants you to decide when two figures are congruent or similar, and to use the ratio of corresponding sides of similar figures to find unknown lengths. Congruence is about being identical; similarity is about being the same shape at a different size. The scale factor is the tool for similarity calculations.

The answer

Congruent figures

Two figures are congruent if they are exactly the same shape and the same size. That means:

  • All pairs of corresponding sides are equal.
  • All pairs of corresponding angles are equal.

A congruent figure can be a rotation, reflection or translation of the other - it just has to match exactly when laid on top.

Similar figures

Two figures are similar if they are the same shape but possibly different sizes. That means:

  • All pairs of corresponding angles are equal.
  • All pairs of corresponding sides are in the same ratio.

So one figure is an enlargement of the other. Every congruent pair is also similar (with scale factor 11), but not every similar pair is congruent.

The scale factor

The scale factor is how many times bigger one figure is than the other. Find it from a pair of corresponding sides:

scale factor=length on the larger figurematching length on the smaller figure\text{scale factor} = \frac{\text{length on the larger figure}}{\text{matching length on the smaller figure}}

Once you have it, multiply (or divide) other sides to find unknown lengths.

Matching corresponding parts

The trick is matching the parts correctly. Corresponding sides lie between corresponding angles. In a clear diagram, the longest side of one figure matches the longest side of the other, and so on.

Examples in context

Example 1. A scale drawing. A floor plan is similar to the real room, drawn at a scale of 1:1001 : 100. A wall measuring 3 cm3\ \text{cm} on the plan corresponds to 300 cm300\ \text{cm} (that is 3 m3\ \text{m}) in real life. Scale drawings are similarity in everyday use, where the scale is the scale factor.

Example 2. Shadows and heights. A person and a tree cast shadows at the same time, forming two similar triangles (the angle of the sun is shared). Comparing the ratio of height to shadow for the person gives the tree's height. Similar triangles let you measure tall objects indirectly.

Try this

  • Cue. Two similar triangles have corresponding sides 5 cm5\ \text{cm} and 15 cm15\ \text{cm}. The scale factor is 155=3\dfrac{15}{5} = 3.
  • Cue. A side of 7 cm7\ \text{cm} is enlarged by scale factor 22. The new length is 7×2=14 cm7 \times 2 = 14\ \text{cm}.
  • Cue. Are two squares always similar? Yes - all squares have equal angles (90∘90^\circ) and sides in the same ratio, so any two squares are similar.

Exam-style practice questions

Practice questions written in the style of SEAB exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Original3 marksTwo triangles are similar. The first has a side of 4 cm4\ \text{cm} that corresponds to a side of 12 cm12\ \text{cm} in the second. Another side of the first triangle is 5 cm5\ \text{cm}. Find the corresponding side in the second triangle.
Show worked answer →

Find the scale factor from the matching pair: 124=3\dfrac{12}{4} = 3.

The second triangle is 33 times larger, so the corresponding side is:

5×3=15 cm5 \times 3 = 15\ \text{cm}.

What markers reward: finding the scale factor from a known pair of corresponding sides, then multiplying the other side by it. Using non-corresponding sides to find the scale factor is the common error.

Original2 marksState the condition for two figures to be congruent, and the condition for two figures to be similar.
Show worked answer →

Congruent figures are exactly the same shape and the same size: all corresponding sides and all corresponding angles are equal.

Similar figures are the same shape but not necessarily the same size: corresponding angles are equal and corresponding sides are in the same ratio.

What markers reward: "same shape and size" with equal sides and angles for congruence, and "same shape, equal angles, sides in proportion" for similarity. The key difference is whether the size is the same.

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