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.
15 questions · official format, shuffled order
1 A triangle has sides 5 cm, 7 cm and 9 cm. A similar triangle has shortest side 15 cm. How long is its longest side?
Show worked solution
Scale factor 3, so 9 × 3 = 27 cm. (21 is the middle side; 31 adds 22 instead of multiplying.)
2 A girl 1.5 m tall casts a 2 m shadow. At the same moment a tree casts a 14 m shadow. How tall is the tree?
Show worked solution
1.5 ÷ 2 = 0.75, and 0.75 × 14 = 10.5 m. (18.7 multiplies by 2 ÷ 1.5 — the ratio upside down.)
3 In triangle ABC, DE is parallel to BC. AD = 4, DB = 6 and DE = 5. How long is BC?
Show worked solution
Scale factor 10 ÷ 4 = 2.5, so BC = 5 × 2.5 = 12.5. (7.5 uses DB ÷ AD = 1.5 as the scale factor.)
4 On a map with scale 1 : 45 000, two towns are 6.4 cm apart. How far apart are they really?
Show worked solution
288 000 cm = 2880 m = 2.88 km.
5 A 2.5 km path is 5 cm long on a map. What is the map’s scale?
Show worked solution
5 cm : 250 000 cm = 1 : 50 000.
6 Two similar rectangles have matching sides in the ratio 3 : 4. The smaller one has area 18 cm2. What is the area of the larger one?
Show worked solution
18 × 16/9 = 32 cm2. (24 scales the area by 4/3, as if it were a length.)
7 Two similar jugs are 10 cm and 15 cm tall. The smaller one holds 400 mL. How much does the larger one hold?
Show worked solution
(1.5)3 = 3.375, and 400 × 3.375 = 1350 mL. (600 scales by 1.5; 900 by 1.52.)
8 Two similar solids have volumes in the ratio 1 : 27. The smaller has surface area 14 cm2. What is the surface area of the larger?
Show worked solution
Lengths 1 : 3, areas 1 : 9, so 14 × 9 = 126 cm2. (42 scales by the length ratio; 378 by the volume ratio.)
9 1 Any two squares are similar.
2 Any two isosceles triangles are similar.
3 Two right-angled triangles that share one other equal angle are similar.
Which statements are correct?
2 Any two isosceles triangles are similar.
3 Two right-angled triangles that share one other equal angle are similar.
Which statements are correct?
Show worked solution
Squares always have the same shape (1 ✓). Isosceles triangles can have different angles (2 ✗). Two equal angles force the third, so the triangles match (3 ✓). 1 and 3.
10 A model car is built to a scale of 1 : 20 and is 22 cm long. How long is the real car?
Show worked solution
440 cm = 4.4 m.
11 A 1 : 10 model of a house needs 0.3 L of paint to cover its outside. How much paint would the real house need?
Show worked solution
0.3 × 100 = 30 L. (3 L scales by 10, as if paint covered length.)
12 A room has a floor area of 45 m2. It is drawn on a plan with scale 1 : 100. What is its area on the plan?
Show worked solution
45 m2 ÷ 10 000 = 0.0045 m2 = 450 cm2. (45 cm2 treats the area scale as 1 : 100 and also forgets the unit change.)
13 In triangle ABC, DE is parallel to BC and AD : DB = 2 : 3. Triangle ADE has area 8 cm2. What is the area of the trapezium DBCE?
Show worked solution
Areas 4 : 25, so ABC = 8 × 25/4 = 50 and the trapezium is 50 − 8 = 42 cm2. (50 is the whole triangle; 12 uses 2 : 3 as the length ratio.)
14 A 6 cm by 10 cm rectangle is similar to a rectangle whose longer side is 25 cm. What is the other rectangle’s shorter side?
Show worked solution
Scale factor 2.5, so 6 × 2.5 = 15 cm. (21 adds 15 to 6 — the difference 25 − 10 — instead of multiplying.)
15 A map has scale 1 : 600 000.
1 1 cm on the map is 6 km on the ground.
2 1 cm2 on the map is 6 km2 on the ground.
3 An 18 km road is 3 cm long on the map.
Which statements are correct?
1 1 cm on the map is 6 km on the ground.
2 1 cm2 on the map is 6 km2 on the ground.
3 An 18 km road is 3 cm long on the map.
Which statements are correct?
Show worked solution
600 000 cm = 6 km (1 ✓). 6 km × 6 km = 36 km2, not 6 (2 ✗). 18 ÷ 6 = 3 cm (3 ✓). 1 and 3.