Step-by-step worked examples and graded practice questions on similarity and congruence — congruence criteria, similar shapes, scale factors, and area and volume ratios.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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Two shapes are congruent if they are exactly the same size and shape — one could be rotated, reflected or translated to fit exactly on top of the other. For triangles, there are four ways to prove congruence:
SSS — all three sides match
SAS — two sides and the angle between them (the included angle) match
ASA — two angles and the side between them (the included side) match
RHS — both triangles are right-angled, with matching hypotenuses and one other matching side
Worked Example 1 — SAS congruence
Triangle ABC has AB = 8cm, AC = 6cm, and the angle at A is 90°. Triangle DEF has DE = 8cm, DF = 6cm, and the angle at D is 90°. Explain why the triangles are congruent.
1
Two pairs of sides match: AB = DE = 8cm, and AC = DF = 6cm.
2
The angle between these two sides also matches: angle A = angle D = 90°.
AnswerCongruent by SAS
Even though the second triangle is rotated, the matching sides and included angle prove the two triangles are congruent (SAS).
Similar shapes
Two shapes are similar if they have exactly the same shape but a different size — all corresponding angles are equal, and all corresponding sides are in the same ratio (the scale factor).
The two triangles have the same angles (matching arcs) and every side of the larger triangle is exactly double the corresponding side of the smaller one — a scale factor of 2.
Finding a missing side
If two shapes are similar, you can find a missing length by working out the scale factor from a pair of known corresponding sides, then applying it to the side you need.
Worked Example 2 — Finding a missing side
Triangle ABC is similar to triangle DEF. AB = 4cm, BC = 5cm, and DE = 8cm. Find the length of EF.
1
AB corresponds to DE, so the scale factor = DE ÷ AB = 8 ÷ 4 = 2
2
BC corresponds to EF, so EF = BC × scale factor = 5 × 2
AnswerEF = 10cm
AB and DE are a known corresponding pair — use them to find the scale factor, then apply it to BC to find EF.
Area and volume scale factors
When two shapes are similar with linear scale factor k, their areas scale by k² and (for similar solids) their volumes scale by k³. This connects directly to enlargement and to volume of similar solids.
Area scale factor = (linear scale factor)²
Volume scale factor = (linear scale factor)³
Worked Example 3 — Area scale factor
Two similar shapes have a linear scale factor of 3. Find the area scale factor.
1
Area scale factor = (linear scale factor)² = 3²
Answer9
Practice questions
Work through each question before checking the answers. Where a question shows two triangles or a real-world scenario, the diagram is required to answer it.
Foundation (Grade 3–5)
Q1The diagram shows two triangles. Are they congruent? If so, state the criterion.Foundation
Diagram for Question 1 (not to scale).
Q2Triangle ABC has AB = 5cm, BC = 6cm, and angle B = 70°. Triangle DEF has DE = 5cm, EF = 6cm, and angle E = 70°. Which congruence criterion proves they are congruent?Foundation
Q3Two triangles have all three pairs of angles equal, but one triangle's sides are twice as long as the other's. Are the triangles congruent, similar, or neither?Foundation
Q4Two right-angled triangles have equal hypotenuses and one other pair of equal sides. Which congruence criterion applies?Foundation
Q5Triangle ABC is similar to triangle DEF. AB = 4cm, BC = 5cm, and DE = 8cm. Find the length of EF.Foundation
Diagram for Question 5 (not to scale).
Higher (Grade 5–7)
Q6Triangle ABC is similar to triangle DEF. AB = 6cm, DE = 9cm, and BC = 8cm. Find the length of EF.Higher
Q7Two similar shapes have a linear scale factor of 3. Find the area scale factor.Higher
Q8Two similar shapes have an area scale factor of 25. Find the linear scale factor.Higher
Q9Two similar solids have a linear scale factor of 2. Find the volume scale factor.Higher
Q10A vertical stick 1.5m tall casts a shadow 2m long. At the same time, a nearby tree casts a shadow 12m long. Find the height of the tree.Higher
Diagram for Question 10 (not to scale). Both triangles (stick + shadow, tree + shadow) are similar right-angled triangles.
Higher — Hard (Grade 8–9)
Q11Two similar solids have a volume ratio of 8:27. Find the linear scale factor.Grade 8–9
Q12Triangle ABC is similar to triangle DEF. AB = 8cm, BC = 6cm, and EF = 15cm. DE = x cm. Find x.Grade 8–9
Diagram for Question 12 (not to scale).
Q13Two similar cylinders have radii 4cm and 10cm. The smaller cylinder has a volume of 96cm³. Find the volume of the larger cylinder.Grade 8–9
Q14The diagram shows two straight lines crossing between a pair of parallel lines, forming two similar triangles. Find the missing length.Grade 8–9
Diagram for Question 14 (not to scale). The two horizontal lines are parallel.
Q15Two similar shapes have an area ratio of 9:16. A side on the smaller shape is 6cm. Find the corresponding side on the larger shape.Grade 8–9
Answers
Foundation (Q1–Q5)
Q1Yes — congruent by SSS (all three sides match: 6cm, 7cm, 9cm)
Q2SAS
Q3Similar (same shape, different size — not congruent)
Q4RHS
Q510cm(scale factor 2, so EF = 5 × 2)
Higher (Q6–Q10)
Q612cm(scale factor 1.5, so EF = 8 × 1.5)
Q79(3²)
Q85(√25)
Q98(2³)
Q109m(1.5 × (12 ÷ 2))
Higher — Hard (Q11–Q15)
Q111.5(∛(27/8) = 3/2)
Q12x = 20(scale factor 15÷6 = 2.5, so DE = 8 × 2.5)
Congruent shapes are identical in size. Similar shapes have the same shape but can be different sizes. Every congruent shape is similar, but not every similar shape is congruent.
Common Mistake 2
Forgetting to square or cube the scale factor for area and volume
A linear scale factor of 2 gives an area scale factor of 4 (2²) and a volume scale factor of 8 (2³) — using the linear scale factor directly for area or volume is a very common error.
Common Mistake 3
Matching up the wrong corresponding sides
Always match sides by the vertices they connect (AB corresponds to DE, not necessarily to DF) — mixing up correspondence gives a completely wrong scale factor.
Common Mistake 4
Forgetting to take a root when reversing area or volume scale factors
Going from an area ratio back to a linear scale factor needs a square root; from a volume ratio, a cube root. Forgetting this step is a common Grade 8–9 error.
Exam tips
💡 Exam Tip 1
State the congruence criterion explicitly
Writing "congruent by SAS" (not just "congruent") shows you've correctly identified which sides and angles match — this is usually worth a method mark on its own.
💡 Exam Tip 2
Write the scale factor before using it
Calculate and state the scale factor as its own step — this makes your working clear and lets you reuse it for multiple missing sides in the same question.
💡 Exam Tip 3
Sketch real-world similarity problems as triangles
Shadow, ladder, and map-scale problems are all similar-triangles problems in disguise — draw the two right-angled triangles before setting up the ratio.
💡 Exam Tip 4
Check whether the question is about length, area, or volume
Re-read the question before squaring or cubing the scale factor — using the wrong power is one of the most common lost marks in this topic.
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