1. 1
   A model-maker produces two similar scale models of a car: a smaller one 8 cm long and a larger one 20 cm long.

   (a)

   Two model cars are geometrically similar. The smaller model is 8 cm long and the larger is 20 cm long. What is the scale factor from the smaller to the larger model?

   [1 mark]

   • A$2.5$
   • B$0.4$
   • C$12$
   • D$1.6$

   (b)

   A wheel on the smaller model has diameter 1.2 cm. Find the diameter of the corresponding wheel on the larger model.

   [1 mark]

   • A$2.5$ cm
   • B$3$ cm
   • C$1.2$ cm
   • D$4.8$ cm

   (c)

   If the smaller model's windscreen has area $4$ cm$^2$, what is the area of the corresponding windscreen on the larger model, using the length ratio of $2.5$?

   [1 mark]

   • A$6.25$ cm$^2$
   • B$10$ cm$^2$
   • C$25$ cm$^2$
   • D$40$ cm$^2$

   Total for question 1: 3 marks

2. 2
   Triangle PQR is similar to triangle STU. PQ = 6 cm corresponds to ST = 9 cm, and QR = 8 cm.

   (a)

   Triangle PQR is similar to triangle STU, with PQ = 6 cm corresponding to ST = 9 cm. If QR = 8 cm, find the length of TU.

   [2 marks]

   (b)

   Given angle PQR = $72°$, state the size of angle STU, giving a reason.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sculptor creates two similar statues. The smaller statue has height 40 cm and volume $2000$ cm$^3$; the larger statue has height 100 cm.

   (a)

   Two solid statues are similar. The smaller statue has a height of 40 cm and a volume of $2000$ cm$^3$. Find the length scale factor from the smaller to the larger statue, given the larger statue has a height of 100 cm.

   [3 marks]

   (b)

   Using the length scale factor found in part (a), calculate the volume of the larger statue.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   Triangles ABC and DEF are being compared for congruence, and a further pair of similar triangles GHI and JKL are used to find an unknown length.

   (a)

   Two triangles, ABC and DEF, have AB = DE, BC = EF, and angle ABC = angle DEF. State the condition (SSS, SAS, ASA or RHS) that proves these triangles are congruent, and explain why this condition applies here.

   [4 marks]

   (b)

   Triangles GHI and JKL are similar, with GH = 5 cm, HI = 7 cm, GI = 9 cm, and JK = 15 cm (corresponding to GH). Find the lengths of KL and JL.

   [4 marks]

   (c)

   Explain, using angle facts, why triangle GHI (with angles $40°$, $60°$ and $80°$) is similar to a third triangle MNO with angles $40°$ and $60°$ given directly, without needing to know all three angles of MNO or any side lengths.

   [5 marks]

   Total for question 4: 13 marks

End of questions