Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 A 12 cm by 8 cm rectangle is similar to a smaller rectangle whose shorter side is 6 cm. Find the scale factor k (large to small) and the smaller rectangle's longer side.
Show worked solution
k = 6 ÷ 8 = \(\dfrac{3}{4}\), and 12 × \(\dfrac{3}{4}\) = 9: k = \(\dfrac{3}{4}\), longer side 9 cm. k = \(\dfrac{1}{2}\) matches the 6 with the 12 (the wrong side); 10 cm subtracts 2 instead of multiplying; k = \(\dfrac{4}{3}\) is upside down and enlarges instead of reducing.
2 Triangle DEF is similar to triangle ZYX, with the equal angles marked. DE = 4 cm, EF = 6 cm, DF = 7 cm and ZY = 10 cm. Find YX and ZX.
Show worked solution
k = ZY ÷ DE = 10 ÷ 4 = 2.5. YX matches EF: 6 × 2.5 = 15 cm. ZX matches DF: 7 × 2.5 = 17.5 cm.
3 At the same moment, a 1.2 m stick casts a 0.8 m shadow and a flagpole casts a 9 m shadow. How tall is the flagpole, in metres?
Show worked solution
Stick: 1.2 ÷ 0.8 = 1.5, so every height is 1.5 times its shadow. Flagpole: 9 × 1.5 = 13.5 m.
Level 2 · Core
4 In triangle ABC, D is on AB and E is on AC, with DE ∥ BC. AD = 3, DB = 6 and AE = 4. Find EC.
Show worked solution
AB = 3 + 6 = 9, so k = 9 ÷ 3 = 3 and AC = 4 × 3 = 12. EC = 12 − 4 = 8. 2 uses DB ÷ AD the wrong way round (4 × 3 ÷ 6); 6 assumes EC is the same as DB; 12 is the whole side AC (forgot to subtract AE).
5 Triangle PQR is similar to triangle STU, with the vertices in that order. PQ = 8 cm, ST = 20 cm and QR = 6 cm. Write the ratio PQ : ST in simplest form, and find TU.
Show worked solution
8 : 20 = 2 : 5 (divide both by 4). So k = \(\dfrac{5}{2}\) = 2.5 and TU = 6 × 2.5 = 15 cm.
6 Mia, who is 1.8 m tall, stands 12 m from a lamp post on flat ground. Her shadow is 3 m long. How tall is the lamp post, in metres?
Show worked solution
The big triangle's base runs from the post to the shadow tip: 12 + 3 = 15 m. Mia: 1.8 ÷ 3 = 0.6. Post: 0.6 × 15 = 9 m. Using 12 m as the base gives 7.2 m, the wrong triangle.
Level 3 · Challenge
7 Three statements:
1 A triangle with angles 50° and 60° is similar to a triangle with angles 50° and 70°.
2 A 4 by 6 rectangle is similar to a 6 by 8 rectangle.
3 Triangle ABC is enlarged with scale factor 3 to give triangle DEF. If DE = 12, then AB = 36.
Which statements are correct?
1 A triangle with angles 50° and 60° is similar to a triangle with angles 50° and 70°.
2 A 4 by 6 rectangle is similar to a 6 by 8 rectangle.
3 Triangle ABC is enlarged with scale factor 3 to give triangle DEF. If DE = 12, then AB = 36.
Which statements are correct?
Show worked solution
The third angles are 70° and 60°, so both triangles have angles 50°, 60°, 70° and are similar (1 ✓). 6 ÷ 4 = 1.5 but 8 ÷ 6 ≈ 1.33, so the rectangles are not similar (2 ✗). DEF is the enlargement, so AB = 12 ÷ 3 = 4, not 36 (3 ✗). 1 only.
8 Triangle ABC has a right angle at A, AB = 12 cm, AC = 9 cm and BC = 15 cm. D is on AB with AD = 6 cm, and E is on AC with ∠ADE = ∠ACB (marked). Find AE and DE.
Show worked solution
Matching angles: A ↔ A, D ↔ C, E ↔ B, so AD ↔ AC, AE ↔ AB, DE ↔ CB. k = AD ÷ AC = 6 ÷ 9 = \(\dfrac{2}{3}\). AE = 12 × \(\dfrac{2}{3}\) = 8 cm and DE = 15 × \(\dfrac{2}{3}\) = 10 cm. Check: 62 + 82 = 102 ✓. Treating DE as parallel to BC (AD ↔ AB) gives 4.5 and 7.5, which is wrong.
9 In triangle ABC, D is on AB and E is on AC, with DE ∥ BC. AD = x cm, DB = 9 cm, DE = 6 cm and BC = 15 cm. Find x.
Show worked solution
x ÷ (x + 9) = 6 ÷ 15 = \(\dfrac{2}{5}\), so 5x = 2(x + 9) = 2x + 18, 3x = 18 and x = 6. Check: AB = 6 + 9 = 15, and AB ÷ AD = 15 ÷ 6 = 2.5 = BC ÷ DE = 15 ÷ 6 ✓.