Similarity and congruence – Year 11 GCSE maths test answers to download as PDF

← Back to the test

1. Answer to Question 1 (4 marks)

   1. \(\dfrac{15}6=\) 2.5 (1) Divide the corresponding sides: the second length divided by the first, \(15\div6\).
   2. \(8\times2.5=\) 20 cm (1) Multiply the other side of the first rectangle by the scale factor 2.5.
   3. \(2.5^2=6.25\), so \(1:6.25\) or \(4:25\) (2) The area scale factor is the square of the length scale factor. Check: \(6\times8=48\) and \(15\times20=300\), and \(\tfrac{300}{48}=6.25\).

   In (c), \(1:6.25\) is the same as \(4:25\) after multiplying both parts by 4.

2. Answer to Question 2 (4 marks)

   1. Equal corresponding angles (parallel lines) and a common angle at \(A\) (1) Because \(DE\) is parallel to \(BC\), angle \(ADE=\) angle \(ABC\) and angle \(AED=\) angle \(ACB\) (corresponding angles), so the triangles have equal angles.
   2. \(AB=8\), scale factor \(\dfrac83\) (1), \(BC=4.5\times\dfrac83\) (1) \(=\) 12 cm (1) \(AB=AD+DB=3+5\), so the scale factor from \(ADE\) to \(ABC\) is \(\tfrac{AB}{AD}=\tfrac83\). Multiply \(DE\) by this factor.

3. Answer to Question 3 (4 marks)

   1. Area factor \(\dfrac{25}{9}\) (1); \(90\times\dfrac{25}9=\) 250 cm² (1) The area scale factor is the square of the length ratio: \(\left(\tfrac53\right)^2\). Multiply because we go from small to large.
   2. Volume factor \(\dfrac{125}{27}\) (1); \(108\times\dfrac{125}{27}=\) 500 cm³ (1) The volume scale factor is the cube of the length ratio: \(\left(\tfrac53\right)^3\).

   A common error is to multiply by the length factor \(\tfrac53\) for both parts, which gives 150 cm² and 180 cm³ and scores no accuracy marks.

4. Answer to Question 4 (4 marks)

   1. Triangles \(AMD\) and \(BMC\): \(AM=BM\) (\(M\) is the midpoint) (1); angle \(A=\) angle \(B=90^\circ\) (1); \(AD=BC\) (opposite sides of a rectangle) (1). So they are congruent by SAS, giving \(MD=MC\) (1).

   Congruent triangles have all corresponding sides equal, so the third sides \(MD\) and \(MC\) are equal. Each statement in the proof must have its reason. The condition used is SAS: two sides and the angle between them. Writing only that the diagram looks symmetrical scores no marks; the proof must name the pair of triangles.

5. Answer to Question 5 (4 marks)

   1. \(30\times12=360\) cm \(=\) 3.6 m (1) Real lengths are 12 times the model lengths. Change 360 cm into metres by dividing by 100.
   2. Volume factor \(12^3=1728\) (1); \(2\times1728=\) 3 456 litres (2) Capacity is a volume, so it scales with the cube of the length scale factor.

   A common error in (b) is to multiply 2 litres by 12, giving 24 litres, or by \(12^2=144\).

Total : 20 marks