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Topics · Congruence & Similarity

Congruence and similarity in DSE Maths Paper 2

This topic is about figures that have the same shape. You need the conditions for two triangles to be congruent or similar, and how lengths, areas and volumes scale between similar figures — most often a triangle cut by a line parallel to one side.

What this topic covers

  • Conditions for congruent and similar triangles
  • Ratios of corresponding sides in similar figures
  • Areas scale by the square of the length ratio, volumes by the cube
  • Lines parallel to one side of a triangle

Read the study guide on this topic

Two sample questions

Sample 1

In the figure, D and F are points on AB, and E and G are points on AC, such that DE FG BC. It is given that AD : DF : FB equals 3 : 4 : 1 and the area of AFG is 392 cm squared. The area of the trapezium DBCE is

ABCDFEG
  1. A320 cm squared .
  2. B440 cm squared .
  3. C448 cm squared .
  4. D512 cm squared .
Show the answer and solution

Answer: B

Worked solution

AD:AF:AB=3:7:8\displaystyle AD : AF : AB = 3 : 7 : 8△ADE\displaystyle \triangle ADE∼△AFG\displaystyle {}\sim \triangle AFG∼△ABC\displaystyle {}\sim \triangle ABC, so their areas are as 32:72:82\displaystyle 3^2 : 7^2 : 8^2△ABC\displaystyle \triangle ABC=392\displaystyle {}= 392×6449\displaystyle {}\times \dfrac{64}{49}=512 cm2\displaystyle {}= 512\text{ cm}^2△ADE\displaystyle \triangle ADE=392\displaystyle {}= 392×949\displaystyle {}\times \dfrac{9}{49}=72 cm2\displaystyle {}= 72\text{ cm}^2DBCE=512−72=440 cm2\displaystyle DBCE = 512 - 72 = 440\text{ cm}^2

Sample 2

In the figure, angle BAC equals 90 degrees and AD BC, where D lies on BC. If AC equals 65 cm and DC equals 25 cm, then BD equals

ABCD
  1. A85 cm .
  2. B144 cm .
  3. C156 cm .
  4. D169 cm .
Show the answer and solution

Answer: B

Worked solution

△ACD∼△BCA\displaystyle \triangle ACD \sim \triangle BCA (equiangular)AC2\displaystyle AC^2=DC\displaystyle {}= DC×BC\displaystyle {}\times BC⇒BC\displaystyle {}\Rightarrow BC=65225\displaystyle {}= \dfrac{65^2}{25}=169\displaystyle {}= 169 cmBD=BC−DC=169−25=144\displaystyle BD = BC - DC = 169 - 25 = 144 cm

Try a whole paper

The free mock is a full 45-question Paper 2 under exam timing, marked the moment you submit, with a report by topic.

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Original material by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.