Mathematics • Year 7 • Unit 3 • Lesson 17
Missing Sides — Real World
Apply similar-triangle methods to maps, scale models and the classic shadow problem. Same sun = similar triangles; same map = scale factor. Get the proportion right and the missing length falls out.
1. Word problems
Each problem needs you to set up a proportion or scale factor, then solve. Show your working — a single answer with no working only earns half marks.
1.1 — Flagpole shadow. A 2 m tall stick casts a 3 m shadow. At the same time, the school flagpole casts a 12 m shadow.
(a) Why are the two triangles (stick + shadow, flagpole + shadow) similar?
(b) Set up a proportion and find the height of the flagpole. 2 marks
1.2 — Map of a national park. A bushwalking map uses a scale of 1 : 50 000. On the map, the trail from the carpark to the waterfall is 6 cm long.
(a) What does "1 : 50 000" mean in plain English?
(b) Find the real trail length in cm, then convert to m, then to km. 3 marks
1.3 — Scale model house. An architect builds a 1 : 24 scale model of a house. The real house is 6 m tall and 9.6 m wide.
(a) What scale factor takes real → model?
(b) Find the height and width of the model, in cm. 3 marks
1.4 — Drawing on a grid. A small triangle drawn on Sara's notebook has sides 3 cm, 4 cm and 5 cm. She wants to redraw it on a poster so the LONGEST side is 30 cm.
(a) Find the scale factor from notebook to poster.
(b) Find the other two side lengths of the poster triangle. 2 marks
1.5 — Photograph blow-up. A photograph is 10 cm wide by x cm tall. When blown up, it becomes 25 cm wide and 35 cm tall. Both photos are similar.
(a) Find the scale factor from small to big.
(b) Find x (the height of the original photo). 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 The shadow problem (1 m stick casting 1.6 m shadow at the same time as a tree casts an 8 m shadow) works because the two triangles formed are similar. In your own words, explain (i) WHY the two triangles are similar (which fact about the sun matters?), (ii) which side of each triangle is the "height" and which is the "shadow", and (iii) why the answer (tree height = 5 m) makes sense compared to a 1 m stick.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Flagpole shadow
(a) Both shadows are cast at the SAME moment, so the sun's rays hit at the SAME angle. Each object forms a right-angled triangle with its shadow, and the two triangles share the sun-angle — making them similar.
(b) Proportion: flagpole / 2 = 12 / 3. So flagpole = 2 × (12 ÷ 3) = 2 × 4 = 8 m.
1.2 — National park map
(a) "1 : 50 000" means every 1 unit on the map represents 50 000 of the same units in real life. So 1 cm on the map = 50 000 cm in real life.
(b) Real length = 6 × 50 000 = 300 000 cm.
Convert to m: 300 000 ÷ 100 = 3000 m.
Convert to km: 3000 ÷ 1000 = 3 km.
1.3 — Scale model house
(a) Real → model: SF = 1/24 (the model is one twenty-fourth of the real).
(b) Convert to cm first. Real: 6 m = 600 cm, 9.6 m = 960 cm.
Model height = 600 ÷ 24 = 25 cm.
Model width = 960 ÷ 24 = 40 cm.
1.4 — Notebook to poster triangle
(a) SF = 30 ÷ 5 = 6.
(b) Other sides = 3 × 6 = 18 cm and 4 × 6 = 24 cm.
1.5 — Photograph blow-up
(a) SF (small → big) = 25 ÷ 10 = 2.5.
(b) Big → small: x = 35 ÷ 2.5 = 14 cm. So the original photo was 10 × 14.
2.1 — Explain your thinking (sample response)
The two triangles are similar because the sun is so far away that its rays reach both the stick and the tree at the same angle. Each object stands straight up (the "height") and casts a shadow along the ground (the "shadow"), forming a right-angled triangle. Because the sun-angle is the same for both and both shadows are on flat ground (both have right angles where the object meets the ground), the two triangles have equal corresponding angles — that's the definition of similar.
Setting up the proportion: tree / stick = tree shadow / stick shadow → tree / 1 = 8 / 1.6 → tree = 5 m. This answer makes sense because the tree's shadow (8 m) is 5 times the stick's shadow (1.6 m), so the tree should be 5 times the stick's height: 1 m × 5 = 5 m. The scale factor of the shadows (5) matches the scale factor of the heights — exactly what we expect from similar triangles.
Marking: 1 for naming "same sun angle"; 1 for identifying height vs shadow correctly; 1 for showing 5 m makes sense; 1 for clear explanation.