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O-Level Mathematics · Secondary 3 and 4

Congruence and Similarity

Two figures are congruent when they are exactly the same shape and size, and similar when one is an enlargement of the other. In O-Level Mathematics you use the tests for congruent and similar triangles, write short proofs with reasons, find unknown lengths from ratios, and apply area and volume scale factors to similar figures, solids and maps. The ideas build on lower-secondary geometry and are used in trigonometry, circle geometry and vector questions where triangles share an angle or lines are parallel.

What the syllabus expects

  • State and use the tests for congruent triangles: SSS, SAS, ASA or AAS, and RHS
  • State and use the tests for similar triangles: two pairs of equal angles, three pairs of sides in the same ratio, and two pairs of sides in the same ratio with the included angles equal
  • Write a proof that two triangles are congruent or similar, giving a reason for each statement
  • Find unknown lengths and angles in similar figures
  • Use the ratio of areas, (l₁/l₂)², and the ratio of volumes, (l₁/l₂)³, for similar figures and solids
  • Use map scales to convert distances and areas

Congruent triangles

Two triangles are congruent if one can be placed exactly on top of the other, so all corresponding sides and angles are equal. You do not need to check all six measurements. Any one of these tests is enough: SSS (three pairs of equal sides), SAS (two pairs of equal sides and the angles between them), ASA or AAS (two pairs of equal angles and one pair of corresponding sides), or RHS (a right angle, equal hypotenuses and one other pair of equal sides). Two cases are not tests. Three equal angles (AAA) only proves the triangles are similar, since one could be larger than the other. Two sides and a non-included angle (SSA) can fit two different triangles. When naming congruent triangles, write the letters in corresponding order, so triangle ABC ≡ triangle PQR means A matches P, B matches Q and C matches R.

Similar triangles and missing lengths

Similar figures have equal corresponding angles and corresponding sides in the same ratio. Triangles are similar if two pairs of angles are equal (the third pair is then equal too), if all three pairs of sides are in the same ratio, or if two pairs of sides are in the same ratio and the angles between them are equal. A very common set-up is a line drawn parallel to one side of a triangle. If DE is parallel to BC, with D on AB and E on AC, then triangle ADE is similar to triangle ABC, because angle A is shared and the corresponding angles ADE and ABC are equal. To find a missing length, match corresponding sides carefully, often by redrawing the two triangles separately in the same orientation, then write a ratio such as DE/BC = AD/AB. Use the whole side AB, not the part DB.

Area and volume ratios

If two figures are similar with lengths in the ratio k, then their areas are in the ratio k² and, for solids, their volumes are in the ratio k³. So if two similar cones have heights 4 cm and 6 cm, the length ratio is 2 : 3, the area ratio is 4 : 9 and the volume ratio is 8 : 27. Surface area, curved surface area and the area of any face all follow the k² rule, while volume, capacity and the mass of solids made of the same material follow the k³ rule. You can also work backwards: if the volumes of two similar solids are in the ratio 64 : 125, take cube roots to get the length ratio 4 : 5, then square for the area ratio 16 : 25. Always check whether the given quantity is a length, an area or a volume before choosing the power.

Map scales

A map scale such as 1 : 50 000 means 1 cm on the map represents 50 000 cm on the ground, which is 500 m or 0.5 km. Distances scale by this ratio directly, so 7 cm on the map represents 3.5 km. Areas scale by the square of the ratio, and it is safest to convert the length first and then square it: 1 cm represents 0.5 km, so 1 cm² represents 0.5² = 0.25 km². A field of area 3 cm² on this map has an actual area of 3 × 0.25 = 0.75 km². Working backwards, an actual area of 2 km² appears as 2 ÷ 0.25 = 8 cm² on the map. To change units, remember that 1 km = 1000 m = 100 000 cm. Writing the line '1 cm represents ... km' before anything else keeps the conversions organised.

Worked example

In triangle ABC, D lies on AB and E lies on AC such that DE is parallel to BC. AD = 4 cm, DB = 6 cm and BC = 15 cm. (a) Show that triangle ADE is similar to triangle ABC. (b) Find DE. (c) Given that the area of triangle ADE is 12 cm², find the area of the trapezium DBCE.

  1. Angle DAE = angle BAC, since the angle at A is common to both triangles.
  2. Angle ADE = angle ABC, since they are corresponding angles with DE parallel to BC. So triangle ADE is similar to triangle ABC, as two pairs of angles are equal.
  3. AB = AD + DB = 4 + 6 = 10 cm, so the length ratio is AD/AB = 4/10 = 2/5.
  4. DE/BC = 2/5, so DE = (2/5) × 15 = 6 cm.
  5. The area ratio is (2/5)² = 4/25, so the area of triangle ABC = 12 × 25/4 = 75 cm².
  6. Area of DBCE = 75 − 12 = 63 cm².

Answer: (b) DE = 6 cm (c) Area of DBCE = 63 cm²

Common mistakes

  • Using part of a side instead of the whole side. With D on AB, the ratio is AD/AB = 4/10, not AD/DB = 4/6. The small triangle is ADE and the large triangle is ABC, so compare their full sides.
  • Applying the length ratio to areas. If lengths are in the ratio 2 : 5, areas are in the ratio 4 : 25, not 2 : 5. Square for area and cube for volume.
  • Mixing up the parts in an area ratio. The ratio 4 : 25 compares triangle ADE with the whole triangle ABC. The trapezium DBCE is the difference, 25 − 4 = 21 parts, not 25 parts.
  • Quoting AAA or SSA as a congruence test. AAA only shows similarity, and SSA is not a valid test because two different triangles can fit the same information.
  • Writing triangle names in the wrong order. Triangle ABC ≡ triangle QPR is a different statement from triangle ABC ≡ triangle PQR. Match the vertices so that equal angles are in the same position.
  • Giving reasons that are too vague. Write 'corresponding angles, DE parallel to BC' or 'common angle', not just 'similar' or 'parallel lines'.

Exam tips

  • Redraw the two triangles side by side in the same orientation and mark the equal angles. Matching corresponding sides becomes much easier.
  • In a proof, write one statement per line with its reason, and finish with the test used, such as 'so triangle ABC ≡ triangle DEF (SAS)'.
  • Before using a ratio, write a line such as 'length ratio 2 : 5, area ratio 4 : 25'. Seeing both ratios together stops you using the wrong one.

Practice questions

1. Two triangles are similar. The smaller has sides 3 cm, 5 cm and 6 cm, and the shortest side of the larger triangle is 4.5 cm. Find the other two sides of the larger triangle.

Show answer

7.5 cm and 9 cm. The scale factor is 4.5 ÷ 3 = 1.5, so the other sides are 5 × 1.5 = 7.5 cm and 6 × 1.5 = 9 cm.

2. A map has a scale of 1 : 25 000. (a) Find the actual distance, in km, represented by 6 cm on the map. (b) A park has an area of 8 cm² on the map. Find its actual area in km².

Show answer

(a) 1.5 km, since 6 × 25 000 = 150 000 cm = 1.5 km. (b) 0.5 km². 1 cm represents 0.25 km, so 1 cm² represents 0.0625 km², and 8 × 0.0625 = 0.5 km².

3. Two similar bottles have heights 10 cm and 15 cm. The smaller bottle holds 400 cm³ and has a surface area of 240 cm². Find the capacity and the surface area of the larger bottle.

Show answer

1350 cm³ and 540 cm². The length ratio is 15/10 = 3/2. Capacity = 400 × (3/2)³ = 400 × 27/8 = 1350 cm³. Surface area = 240 × (3/2)² = 240 × 9/4 = 540 cm².

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