← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 4 Geometry · Chapter 23 · pp. 293–309

Similarity and congruence

Similar shapes are the same shape but can be different sizes: one is an enlargement of the other. Congruent shapes are the same shape and the same size. One number, the scale factor k, tells you how lengths change, and from it how areas (k²) and volumes (k³) change.

Learning objectives

  1. Decide whether two shapes are similar, and find the scale factor.
  2. Calculate missing lengths in similar shapes, including triangles cut by a parallel line.
  3. Use area factor = k² and volume factor = k³ for similar shapes and solids. (Extended)
  4. Decide whether two shapes are congruent.
  5. Use the SSS, SAS, ASA, AAS and RHS tests for congruent triangles.

Book exercises: 23.1 to 23.6 and Student assessment 1 (pp. 293–309).

Similar shapes

🔑 Key ideas

  • Two polygons are similar if (a) they are equi-angular (matching angles are equal) and (b) matching sides are in proportion.
  • For triangles, equal angles is enough: it forces the sides into proportion, and the other way round.
  • The scale factor k is the ratio of matching lengths: big ÷ small to enlarge, small ÷ big to reduce.
  • Heights scale by k as well: = = = = k.
A B C 40° 35° 105° c b a h P Q R 40° 35° 105° r q p H k = p/a = q/b = r/c every pair of matching lengths gives the same k
∆ABC and ∆PQR have the same three angles, so they are similar. Each side of ∆PQR is k times the matching side of ∆ABC, and so is the height.

📘 Example 1 · ∆ABC is similar to ∆PQR, with AB = 4 cm, BC = 6 cm, AC = 5 cm and PQ = 10 cm. Find QR and PR.

  1. Match the sides: AB ↔ PQ, BC ↔ QR, AC ↔ PR.
  2. Scale factor from a known pair: k = PQ ÷ AB = 10 ÷ 4 = 2.5.
  3. QR = BC × k = 6 × 2.5 = 15 cm.
  4. PR = AC × k = 5 × 2.5 = 12.5 cm.
  5. Answer: QR = 15 cm, PR = 12.5 cm. Check: 15 ÷ 6 = 12.5 ÷ 5 = 2.5 ✔

A line parallel to one side

A line drawn parallel to one side of a triangle cuts off a smaller triangle with the same angles (corresponding angles are equal), so the two triangles are similar.

A D E B C 3 6 4 BC = ? Small ∆ADE AD = 3, DE = 4 Big ∆ABC AB = 3 + 6 = 9 (the whole side) k = 9 ÷ 3 = 3 BC = DE × 3 = 12
DE is parallel to BC. Pull the two triangles apart in your head: the big side is the whole of AB, not just the 6 below D.

📘 Example 2 · In ∆ABC, DE ∥ BC with AD = 3 cm, DB = 6 cm and DE = 4 cm. Find BC.

  1. ∆ADE and ∆ABC share angle A, and DE ∥ BC makes the other angles equal, so they are similar.
  2. Use the whole side for the big triangle: AB = AD + DB = 3 + 6 = 9 cm.
  3. Scale factor: k = AB ÷ AD = 9 ÷ 3 = 3.
  4. BC = DE × k = 4 × 3 = 12 cm.

📘 Example 3 · Two similar triangles: the big one has sides 12 cm and 9 cm; the matching 12 cm side on the small one is 8 cm. Find the side matching 9 cm.

  1. Going from big to small, so the scale factor reduces: k = 8 ÷ 12 = .
  2. Multiply the matching side: 9 × = 6.
  3. Answer: 6 cm. Check: 8 : 12 = 6 : 9 = 2 : 3 ✔

✏️ Try it

  1. A 3 cm by 5 cm rectangle is similar to a rectangle whose short side is 6 cm. Find its long side.
  2. A triangle with sides 5, 7 and 8 cm is enlarged so the 5 cm side becomes 15 cm. Find the other two sides.
  3. Are all isosceles triangles similar? Give a reason.
Answers

1. k = 2, so 10 cm · 2. k = 3: 21 cm and 24 cm · 3. No: their angles can differ (for example 50°, 50°, 80° and 70°, 70°, 40°).

Next: area of similar shapes →

Area of similar shapes Extended

🔑 Key ideas

  • Area factor = (scale factor)² = k².
  • Why: area is length × length, and both lengths are multiplied by k. For triangles, = k × k = k².
  • Going backwards: length factor k = √(area factor).
  • A percentage change in length is a multiplier: +10% means k = 1.1, −15% means k = 0.85.
k = 1 area 1 k = 2 area 4 k = 3 area 9 Area factor = k² sides × 2 → 2 × 2 = 4 squares sides × 3 → 3 × 3 = 9 squares
Double every side of a square and four copies of the original fit inside it. Treble the sides and nine fit.

📘 Example 4 · Two similar triangles have bases 4 cm and 12 cm. The small one has area 10 cm². Find the area of the big one.

  1. Length scale factor: k = 12 ÷ 4 = 3.
  2. Area factor: k² = 3² = 9.
  3. Big area = 10 × 9 = 90 cm².

📘 Example 5 · Two similar shapes have areas 20 cm² and 45 cm². A side of the small one is 6 cm. Find the matching side of the big one.

  1. Area factor: 45 ÷ 20 = 2.25.
  2. Length factor is the square root: k = √2.25 = 1.5.
  3. Matching side = 6 × 1.5 = 9 cm.
  4. Check: (9 ÷ 6)² = 1.5² = 2.25 = 45 ÷ 20 ✔

📘 Example 6 · The sides of a 5 cm square are increased by 20%. Find the new area.

  1. A 20% increase is a multiplier: k = 1.2.
  2. Area factor: k² = 1.2² = 1.44.
  3. Old area = 5² = 25 cm², so new area = 25 × 1.44 = 36 cm².
  4. Check: new side 5 × 1.2 = 6 cm, and 6² = 36 ✔. The area went up 44%, not 20%.

✏️ Try it

  1. A shape of area 3 cm² is enlarged by scale factor 4. Find the new area.
  2. Two similar shapes have areas 16 cm² and 100 cm². Find the length scale factor.
  3. The sides of a shape are reduced by 10%. By what percentage does its area fall?
Answers

1. 3 × 4² = 48 cm² · 2. √(100 ÷ 16) = √6.25 = 2.5 · 3. 0.9² = 0.81, so the area falls by 19%

Next: volume of similar shapes →

Area and volume of similar shapes Extended

🔑 Key ideas

  • Volume factor = (scale factor)³ = k³. Capacity and mass (same material) follow the volume.
  • Surface area of similar solids still scales by k².
  • Going backwards: k = ∛(volume factor) or k = √(area factor).
  • Always go back to k first, then out to the factor you need.
k = 1: 1 cube k = 2: 2 × 2 × 2 = 8 Measure Factor k = 2 Length × k × 2 Area × k² × 4 Volume × k³ × 8
Double every edge of a cube and eight of the small cubes fill the big one: volume factor 2³ = 8.

📘 Example 7 · Two similar cylinders have heights 6 cm and 9 cm. The small one holds 80 cm³. Find the volume of the big one.

  1. Length scale factor: k = 9 ÷ 6 = 1.5.
  2. Volume factor: k³ = 1.5³ = 3.375.
  3. Big volume = 80 × 3.375 = 270 cm³.
Volume ratio 135 ÷ 40 = 3.375 Cube root k = ∛3.375 = 1.5 Square k² = 1.5² = 2.25 Area 48 × 2.25 = 108
From a volume ratio to an area: back to k with a cube root, then square it. (This is the same as raising 3.375 to the power ⅔.)

📘 Example 8 · Two similar bottles have volumes 40 cm³ and 135 cm³. The small one has surface area 48 cm². Find the surface area of the big one.

  1. Volume factor: 135 ÷ 40 = 3.375.
  2. Back to the length factor with a cube root: k = ∛3.375 = 1.5.
  3. Area factor: k² = 1.5² = 2.25.
  4. Big surface area = 48 × 2.25 = 108 cm².

📘 Example 9 · A map has scale 1 : 50 000. A lake covers 6 cm² on the map. Find its real area in km².

  1. 1 cm on the map is 50 000 cm in real life.
  2. Convert: 50 000 cm = 500 m = 0.5 km, so 1 cm ↔ 0.5 km.
  3. Square it for area: 1 cm² ↔ 0.5 × 0.5 = 0.25 km².
  4. Real area = 6 × 0.25 = 1.5 km².

🧪 Try it: scale factor calculator

    ✏️ Try it

    1. A solid of volume 5 cm³ is enlarged by scale factor 3. Find the new volume.
    2. Two similar jugs hold 8 litres and 27 litres. Find the length factor and the area factor.
    3. A map has scale 1 : 20 000. A park covers 4 cm² on the map. Find its real area in km².
    Answers

    1. 5 × 3³ = 135 cm³ · 2. k = ∛(27 ÷ 8) = 1.5, area factor 1.5² = 2.25 · 3. 1 cm ↔ 0.2 km, 1 cm² ↔ 0.04 km², so 4 × 0.04 = 0.16 km²

    Next: congruent shapes →

    Congruent shapes

    🔑 Key ideas

    • Two shapes are congruent if matching sides are the same length and matching angles are the same size: exactly the same size and shape.
    • A congruent copy can be turned (rotated) or flipped (reflected). Moving it does not change it.
    • Congruent shapes are always similar (k = 1). Similar shapes are not necessarily congruent.
    • Name congruent shapes in matching order: ABCD ≅ EFGH means AB = EF, angle A = angle E, and so on.
    X the original Y half-turn of X: congruent Z smaller copy: similar only
    Y is X turned through 180°, so it fits exactly on top of X: congruent. Z has the same angles but shorter sides: similar, not congruent.
    Similar same shape, any size Congruent same shape and same size
    Every congruent pair sits inside the similar set: congruent is the special case where the scale factor is 1.

    📘 Example 10 (book) · ∆ABC and ∆DEF are congruent. ∆ABC has angles of 55° and 40° at B and C, and DE = 6 cm. Find angle FDE and the length AB.

    1. Match the vertices in order: A ↔ D, B ↔ E, C ↔ F.
    2. Angle FDE matches angle CAB.
    3. Angle CAB = 180° − 40° − 55° = 85°, so angle FDE = 85°.
    4. AB matches DE, so AB = 6 cm.

    📘 Example 11 · Quadrilateral (0, 0), (4, 0), (4, 2), (0, 3) is reflected in the y-axis. Is the image congruent? Give its vertices.

    1. A reflection keeps every length and angle, so the image is congruent.
    2. Reflecting in the y-axis changes (x, y) to (−x, y).
    3. Image: (0, 0), (−4, 0), (−4, 2), (0, 3).
    4. Check one side: (4, 0) to (4, 2) is 2 long, and (−4, 0) to (−4, 2) is 2 long ✔

    ✏️ Try it

    1. A triangle is enlarged by scale factor 2. Is the image congruent to it?
    2. Kite PQRS ≅ kite WXYZ. QR = 8 cm and angle Q = 100°. Find XY and angle X.
    3. Two triangles are congruent. One has angles 50° and 60°. What are the angles of the other?
    Answers

    1. No: it is similar but twice the size · 2. XY = 8 cm, angle X = 100° · 3. 50°, 60° and 70°

    Next: congruent triangles →

    Congruent triangles

    You do not need all six sides and angles to prove two triangles congruent. Any one of these five tests is enough:

    TestWhat must match
    SSSAll three sides.
    SASTwo sides and the angle between them.
    ASATwo angles and the side between them.
    AASTwo angles and one other matching side.
    RHSA right angle, the hypotenuse and one other side.
    SSS 3 sides equal SAS 2 sides + angle between ASA 2 angles + side between AAS 2 angles + another side RHS right angle, hyp, side Not a test AAA, or SSA
    Thick sides and arcs show what has to match. The last card is the trap: three angles only prove similarity, and two sides with an angle not between them can give two different triangles.

    📘 Example 12 (book) · Two triangles both have sides 3.5 cm, 4 cm and 5 cm. Are they definitely congruent?

    1. Compare the sides: 3.5 = 3.5, 4 = 4, 5 = 5.
    2. All three matching sides are equal, which is the SSS test.
    3. Answer: yes, congruent (SSS).
    40° B A C₁ C₂ 6 cm 4 cm 4 cm Two sides + an angle NOT between them → two different triangles, ABC₁ and ABC₂ so SSA is not a test
    An arc of radius 4 cm centred at A cuts the base line twice. Both C₁ and C₂ fit the same 40°, 6 cm and 4 cm.

    📘 Example 13 (book) · Two triangles both have sides 6 cm and 4 cm and an angle of 40°, which is not between the two sides. Are they definitely congruent?

    1. Two sides match, but the 40° angle is not between them, so this is not SAS.
    2. Draw AB = 6 cm with the 40° angle at B. An arc of radius 4 cm centred at A meets the base line at two points, C₁ and C₂.
    3. So two different triangles fit the facts.
    4. Answer: not definitely congruent.

    📘 Example 14 · Two right-angled triangles each have a hypotenuse of 13 cm and a side of 5 cm. Are they congruent? Find the third side.

    1. Right angle, hypotenuse and one other side all match: the RHS test.
    2. So the triangles are congruent (RHS).
    3. Third side by Pythagoras: √(13² − 5²) = √(169 − 25) = √144 = 12 cm in both.

    ✏️ Try it

    1. Two triangles have sides 6 cm and 8 cm with an angle of 50° between them. Congruent? Which test?
    2. Two triangles both have angles 40°, 60° and 80°. Are they definitely congruent?
    3. Two right-angled triangles have hypotenuse 10 cm and a side of 6 cm. Which test, and how long is the third side?
    Answers

    1. Yes, SAS · 2. No: they are similar, but could be different sizes (AAA is not a test) · 3. RHS; √(100 − 36) = 8 cm

    Next: summary and quiz →

    Chapter summary

    • Similar: equal angles and sides in proportion. For triangles, equal angles is enough.
    • Scale factor k = matching length ÷ matching length. Lengths and heights × k.
    • A line parallel to one side of a triangle cuts off a similar triangle. Use whole sides, not pieces.
    • Area factor = k², volume factor = k³. Work back to k with √ or ∛ first. (Extended)
    • Map scale 1 : n: lengths × n, areas × n². Convert units before squaring.
    • Congruent: same shape and same size. Reflections and rotations are congruent.
    • Congruent triangle tests: SSS, SAS, ASA, AAS, RHS. AAA and SSA are not tests.

    📄 Practice questions

    The book's Exercises 23.1 to 23.6 and Student assessment 1 are in the printable assignment for this chapter.

    Mini-quiz

    1. Two polygons are similar when they have…
    2. A 3, 4, 5 cm triangle is enlarged so the 3 cm side becomes 12 cm. How long is the side that was 5 cm?
    3. The lengths of two similar shapes are in the ratio 1 : 3. What is the ratio of their areas?
    4. Two similar solids have length scale factor 2. The small one has volume 15 cm³. What is the big one’s volume?
    5. Two similar shapes have areas 25 cm² and 100 cm². What is the length scale factor?
    6. Two similar jars hold 8 litres and 27 litres. What is the ratio of their heights?
    7. The sides of a square are increased by 50%. By what percentage does its area increase?
    8. Which pair of shapes is always similar?
    9. Which of these is NOT a test for congruent triangles?
    10. Two triangles are congruent. One has angles of 45° and 75°. What is the third angle of the other?
    11. Two right-angled triangles both have a 13 cm hypotenuse and a 5 cm side. Which test proves them congruent?
    12. A map has scale 1 : 1000. A field covers 5 cm² on the map. What is its real area?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 23 (Similarity and congruence). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.