Corrigé
8.4 Scale and Similarity
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | A map has scale 1 cm : 200 m. Find the real distance for 4 cm on the map. | 800 m |
| 2 | A scale is 1 : 50. A real length of 25 m is represented by what length on the drawing? | 50 cm |
| 3 | A scale model is 1 : 18. A model car is 25 cm long. Find the length of the real car in metres. | 4.5 m |
| 4 | Two figures are similar with linear scale factor 3. The smaller has length 7 cm. Find the length on the larger figure. | 21 cm |
| 5 | Two similar triangles. The smaller has shortest side 6 cm; the larger has shortest side 18 cm. Find the linear scale factor. | 3 |
| 6 | On a map, 2 cm represents 1 km. Find the real distance for 8 cm. | 4 km |
| 7 | Are these triangles definitely similar? They have angles 50°, 60°, 70° and 50°, 60°, 70°. Justify. | Yes — equal angles → similar (AAA). |
| 8 | A rectangle on a 1 : 50 plan measures 6 cm by 4 cm. Find the real dimensions in metres. | 3 m by 2 m |
| 9 | A scale drawing uses 1 cm : 5 m. The real garden is 35 m wide. Find the drawing width. | 7 cm |
| 10 | Two figures are similar. State two properties of similar figures (give a sentence each). | Corresponding angles are equal; corresponding sides are in the same ratio. |
| 11 | On a map with scale 3 cm : 600 m, a rectangle measures 6 cm by 9 cm. What are its real dimensions in metres? | 1200 m by 1800 m |
| 12 | Two similar triangles have a side ratio of 3 cm : 6 cm corresponding sides. The smaller has another side 5 cm. Find the corresponding side of the larger triangle. | 10 cm |
| 13 | A drawing is at 1 : 25. Two parallel lines on the drawing are 4 cm apart. Find the real distance. | 100 cm = 1 m |
| 14 | Two triangles are similar. The smaller has sides 3, 4, 5 cm. The largest side of the larger triangle is 25 cm. Find the other two sides. | 15 cm and 20 cm |
| 15 | A photo is enlarged by a linear scale factor of 1.5. The original is 10 cm by 15 cm. Find the new dimensions. | 15 cm by 22.5 cm |
| 16 | Convert the scale "1 cm : 5 m" to the form "1 : k". | 1 : 500 |
| 17 | Convert the scale "1 : 50 000" to the form "1 cm : ? km". | 1 cm : 0.5 km |
| 18 | A model dinosaur is 1 : 25 of life-size. Its leg in the model is 8 cm. Find the real leg length in m. | 2 m |
| 19 | Two similar rectangles have lengths 4 cm and 6 cm. Their widths are 3 cm and $w$ cm. Find $w$. | $w = 4.5$ cm |
| 20 | A 30 m tall flagpole casts a 12 m shadow. A nearby tree casts a 5 m shadow. Find the tree height (using similar triangles). | 12.5 m |
| 21 | Two triangles are similar. The smaller has sides 3 cm and 4 cm; the larger has corresponding sides 6 cm and $x$ cm. Find $x$. | $x = 8$ cm |
| 22 | On a floor plan at 1 cm : 50 cm, a room measures 8 cm by 6 cm on the plan. (a) Find the real dimensions. (b) Find the real floor area in $\text{m}^2$. | (a) 400 cm by 300 cm (4 m by 3 m) (b) 12 m² |
| 23 | Two similar shapes have areas in the ratio 9 : 25. Find the linear scale factor. | 3 : 5 |
| 24 | Two similar rectangles have a linear scale factor of 2. The smaller has area 30 cm². Find the area of the larger rectangle. | 120 cm² |
| 25 | A map has scale 1 : 25 000. A path on the map is 6 cm long. Convert the real path length to (a) m, (b) km. | (a) 1500 m (b) 1.5 km |
| 26 | A flagpole and a nearby tree both cast shadows in sunlight. The pole is 12 m tall and casts a 4 m shadow. The tree casts a 6 m shadow. Find the tree height. Why are the triangles similar? | 18 m; similar because both triangles have a vertical line, a horizontal shadow, and the same sun angle. |
| 27 | A drawing at 1 : 200 shows a circle of diameter 5 cm. Find (a) the real diameter; (b) the real area of the circle. | (a) 10 m (b) ≈ 78.5 m² |
| 28 | A triangle on a 1 : 50 plan has a base of 8 cm and height 6 cm. (a) Find the real base and height. (b) Find the real area in m². | (a) 4 m by 3 m (b) 6 m² |
| 29 | A picture frame is similar to another with linear scale factor $\tfrac{3}{4}$. The smaller (with scale factor $\tfrac{3}{4}$) has area 36 cm². Find the area of the larger. | 64 cm² |
| 30 | On the same map (scale 1 : 50 000), three towns A, B, C are at positions A(0, 0), B(12 cm, 0), C(0, 9 cm). (a) Find the real distance AB and the real distance AC. (b) Find the real distance BC. | (a) AB = 6 km, AC = 4.5 km (b) BC = 7.5 km |
| 31 | Rectangle R1 has real dimensions 1200 m by 1800 m. Rectangle R2 has real dimensions 600 m by 900 m. (a) Find the area scale factor $\text{area}(R1):\text{area}(R2)$ in simplest form. (b) Is this the square of the length scale factor? Justify. | (a) $4:1$ (b) Yes; length ratio $2:1$, area ratio $= 4:1$ |
| 32 | Two similar circles have radii a and b, and circumferences 9.4 cm and c cm. (a) Express c in terms of a and b. (b) Show that their areas are in ratio $a^2 : b^2$. | (a) $c = 9.4 b/a$ (b) Area $= \pi r^2$, ratio $a^2:b^2$. |
| 33 | Two similar shapes have linear scale factor $k$. (a) Show that their area ratio is $k^2$. (b) Two similar glaciers have areas 2.16 km² and 0.24 km². Find the linear scale factor. | (a) See working (b) $k = 3$ |
| 34 | A photograph is to be enlarged so the area is doubled. Find the linear scale factor required, exact and to 3 d.p. | $\sqrt{2} \approx 1.414$ |
| 35 | A 1 : 200 scale model of a building uses 12 m² of card. Find the surface area of the real building in m². | 480 000 m² |
| 36 | Triangles ABC and DEF are similar with $A \leftrightarrow D$ etc. Given $AB = 6$, $AC = 8$, $DE = 9$. Find $DF$. | 12 |
| 37 | A 1 : 1000 model city has buildings that take up 50 cm² of model area in total. Find the real area in km². | 0.05 km² |
| 38 | A triangle has vertices $(0,0)$, $(6, 0)$, $(0, 4)$. Enlarge by factor 2 from the origin. Give the image vertices, and check that the area is multiplied by 4. | $(0,0)$, $(12, 0)$, $(0, 8)$; old area 12, new 48 = 12 × 4 ✓ |
| 39 | The map of a city has scale 1 : 12 500. On the map, a park has area 36 cm². Find the real area of the park in m² and in km². | 5.625 × 10⁵ m² ≈ 0.5625 km² |
| 40 | Two similar buildings: the smaller has volume 8 m³ and surface area 24 m². The larger has linear scale factor 3 relative to the smaller. Find (a) the larger volume, (b) the larger surface area. | (a) 216 m³ (b) 216 m² |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | A map has scale 1 cm : 150 m. Find the real distance for 6 cm on the map. | 900 m |
| 2 | A scale is 1 : 100. A real length of 40 m is represented by what length on the drawing? | 40 cm |
| 3 | A scale model is 1 : 24. A model car is 20 cm long. Find the length of the real car in metres. | 4.8 m |
| 4 | Two figures are similar with linear scale factor 2.5. The smaller has length 6 cm. Find the length on the larger figure. | 15 cm |
| 5 | Two similar triangles. The smaller has shortest side 6 cm; the larger has shortest side 18 cm. Find the linear scale factor. | 4 |
| 6 | On a map, 2 cm represents 1 km. Find the real distance for 14 cm. | 7 km |
| 7 | Are these triangles definitely similar? They have angles 50°, 60°, 70° and 50°, 60°, 70°. Justify. | Yes — AAA. |
| 8 | A rectangle on a 1 : 50 plan measures 6 cm by 4 cm. Find the real dimensions in metres. | 5 m by 3 m |
| 9 | A scale drawing uses 1 cm : 5 m. The real garden is 35 m wide. Find the drawing width. | 8 cm |
| 10 | Two figures are similar. State two properties of similar figures (give a sentence each). | Same. |
| 11 | On a map with scale 3 cm : 600 m, a rectangle measures 4 cm by 7 cm. What are its real dimensions in metres? | 800 m by 1400 m |
| 12 | Two similar triangles have a side ratio of 4 cm : 10 cm corresponding sides. The smaller has another side 7 cm. Find the corresponding side of the larger triangle. | 17.5 cm |
| 13 | A drawing is at 1 : 25. Two parallel lines on the drawing are 4 cm apart. Find the real distance. | 200 cm = 2 m |
| 14 | Two triangles are similar. The smaller has sides 3, 4, 5 cm. The largest side of the larger triangle is 25 cm. Find the other two sides. | 15 cm and 36 cm |
| 15 | A photo is enlarged by a linear scale factor of 1.5. The original is 10 cm by 15 cm. Find the new dimensions. | 20 cm by 30 cm |
| 16 | Convert the scale "1 cm : 5 m" to the form "1 : k". | 1 : 250 000 |
| 17 | Convert the scale "1 : 50 000" to the form "1 cm : ? km". | 1 cm : 0.25 km |
| 18 | A model dinosaur is 1 : 25 of life-size. Its leg in the model is 8 cm. Find the real leg length in m. | 1.8 m |
| 19 | Two similar rectangles have lengths 4 cm and 6 cm. Their widths are 3 cm and $w$ cm. Find $w$. | $w = 9.6$ cm |
| 20 | A 30 m tall flagpole casts a 12 m shadow. A nearby tree casts a 5 m shadow. Find the tree height (using similar triangles). | 15 m |
| 21 | Two triangles are similar. The smaller has sides 4 cm and 5 cm; the larger has corresponding sides 12 cm and $x$ cm. Find $x$. | $x = 15$ cm |
| 22 | On a floor plan at 1 cm : 100 cm, a room measures 5 cm by 4 cm on the plan. (a) Find the real dimensions. (b) Find the real floor area in $\text{m}^2$. | (a) 500 cm by 400 cm (5 m by 4 m) (b) 20 m² |
| 23 | Two similar shapes have areas in the ratio 9 : 25. Find the linear scale factor. | 4 : 7 |
| 24 | Two similar rectangles have a linear scale factor of 2. The smaller has area 30 cm². Find the area of the larger rectangle. | 108 cm² |
| 25 | A map has scale 1 : 25 000. A path on the map is 6 cm long. Convert the real path length to (a) m, (b) km. | (a) 5000 m (b) 5 km |
| 26 | A flagpole and a nearby tree both cast shadows in sunlight. The pole is 12 m tall and casts a 4 m shadow. The tree casts a 6 m shadow. Find the tree height. Why are the triangles similar? | 32 m; same reasoning. |
| 27 | A drawing at 1 : 200 shows a circle of diameter 5 cm. Find (a) the real diameter; (b) the real area of the circle. | (a) 4 m (b) ≈ 12.57 m² |
| 28 | A triangle on a 1 : 50 plan has a base of 8 cm and height 6 cm. (a) Find the real base and height. (b) Find the real area in m². | (a) 5 m by 4 m (b) 10 m² |
| 29 | A picture frame is similar to another with linear scale factor $\tfrac{3}{4}$. The smaller (with scale factor $\tfrac{3}{4}$) has area 36 cm². Find the area of the larger. | 72 cm² |
| 30 | On the same map (scale 1 : 50 000), three towns A, B, C are at positions A(0, 0), B(12 cm, 0), C(0, 9 cm). (a) Find the real distance AB and the real distance AC. (b) Find the real distance BC. | Same approach. |
| 31 | Rectangle R1 has real dimensions 1000 m by 1500 m. Rectangle R2 has real dimensions 400 m by 600 m. (a) Find the area scale factor $\text{area}(R1):\text{area}(R2)$ in simplest form. (b) Is this the square of the length scale factor? Justify. | (a) $25:4$ (b) Yes; length ratio $5:2$, area ratio $25:4$ |
| 32 | Two similar circles have radii a and b, and circumferences 9.4 cm and c cm. (a) Express c in terms of a and b. (b) Show that their areas are in ratio $a^2 : b^2$. | See Pack A. |
| 33 | Two similar shapes have linear scale factor $k$. (a) Show that their area ratio is $k^2$. (b) Two similar glaciers have areas 2.16 km² and 0.24 km². Find the linear scale factor. | See Pack A. |
| 34 | A photograph is to be enlarged so the area is doubled. Find the linear scale factor required, exact and to 3 d.p. | $\sqrt{3} \approx 1.732$ |
| 35 | A 1 : 200 scale model of a building uses 12 m² of card. Find the surface area of the real building in m². | 80 000 m² |
| 36 | Triangles ABC and DEF are similar with $A \leftrightarrow D$ etc. Given $AB = 6$, $AC = 8$, $DE = 9$. Find $DF$. | 36 |
| 37 | A 1 : 1000 model city has buildings that take up 50 cm² of model area in total. Find the real area in km². | 0.0075 km² |
| 38 | A triangle has vertices $(0,0)$, $(6, 0)$, $(0, 4)$. Enlarge by factor 2 from the origin. Give the image vertices, and check that the area is multiplied by 4. | $(0,0)$, $(18, 0)$, $(0, 12)$; old 12, new 108 = 12 × 9 ✓ |
| 39 | The map of a city has scale 1 : 12 500. On the map, a park has area 36 cm². Find the real area of the park in m² and in km². | 5 × 10⁵ m² = 0.5 km² |
| 40 | Two similar buildings: the smaller has volume 8 m³ and surface area 24 m². The larger has linear scale factor 3 relative to the smaller. Find (a) the larger volume, (b) the larger surface area. | (a) 64 m³ (b) 96 m² |
Problèmes — Solutions détaillées
**Glacial scale (department November 2025).** On a 1960 map with scale $3$ cm : $600$ m, the rectangle outlining a glacier measures 6 cm by 9 cm. On a 2010 map with scale $5$ cm : $500$ m, the rectangle outlining the same glacier measures 4 cm by 6 cm. (a) Find the real dimensions of each rectangle in metres. (b) Find the real area of each rectangle in square kilometres. (c) Find the ratio Area(R1) : Area(R2) in simplest form. (d) Find the linear scale factor between R1 and R2. (e) Use a linear extrapolation to predict whether the glacier disappears by 2035.
(a) R1 1200 × 1800 m, R2 400 × 600 m (b) R1 2.16 km², R2 0.24 km² (c) 9 : 1 (d) Linear sf 3 (e) ≈ 2016 — earlier than 2035, so the prediction is plausible.
**Floor plan (1 : 50).** A floor plan at 1 cm : 50 cm shows a room measuring 8 cm × 6 cm. (a) Find the real dimensions. (b) Find the real floor area in m². (c) The plan also shows a circular table (radius 0.4 cm on the plan). Find its real radius and area. (d) Show that the plan-area : real-area ratio is $1 : 2500$.
(a) 4 m × 3 m (b) 12 m² (c) Real radius 0.2 m → area ≈ 0.126 m² (d) See working
**Shadow stick.** A 1.5 m vertical stick casts a 1 m shadow at the same time of day that a tree casts a 7 m shadow. (a) Why are the shadow triangles similar? (b) Find the tree height. (c) The shadow of a flagpole is measured later in the day at 5 m, when the stick now casts a 2 m shadow. Find the flagpole height.
(a) Same sun-angle and both vertical → AAA (b) 10.5 m (c) 3.75 m
**Similar pictures.** A photograph is 10 cm × 15 cm. A larger printed copy is similar and has area 600 cm². (a) Find the linear scale factor. (b) Find the larger picture's dimensions. (c) If the smaller picture costs 4 chf and price is proportional to area, find the cost of the larger picture.
(a) 2 (b) 20 cm × 30 cm (c) 16 chf
**Scale-model car.** A scale model of a car is 1 : 24. The real car is 4.8 m long, 1.8 m wide and 1.5 m tall. (a) Find the model's dimensions in cm. (b) Find the surface-area ratio (model : real) and the model's surface area if the real is 36 m². (c) Find the volume ratio (model : real) and the model's volume if the real has interior volume 12 m³.
(a) 20 cm × 7.5 cm × 6.25 cm (b) Area ratio $1 : 576$; model SA $= 625$ cm² (c) Volume ratio $1 : 13824$; model volume $\approx 868$ cm³
**Designing a school garden.** A scale drawing of a school garden uses 1 cm : 2 m. (a) On the drawing, a flowerbed is a triangle with vertices (1 cm, 1 cm), (5 cm, 1 cm), (1 cm, 4 cm). Find the real vertices in metres (relative to the same origin). (b) Find the real area of the flowerbed. (c) A path of width 0.5 m surrounds the flowerbed on its three straight sides. Find the path width on the drawing.
(a) (2, 2), (10, 2), (2, 8) (m) (b) 24 m² (c) 0.25 cm
**Similar shapes investigation.** Two similar shapes have areas 50 cm² and 200 cm². (a) Find the linear scale factor between them. (b) The smaller has perimeter 30 cm. Find the larger perimeter. (c) The smaller has 3 cm of trim around its border. Find the trim length on the larger shape.
(a) 2 (b) 60 cm (c) 6 cm
**Map of Geneva.** On a 1 : 50 000 map, Cornavin station is 12 cm from UN headquarters along a straight line. (a) Find the real distance in metres and kilometres. (b) On a different map at 1 : 25 000, what would the same distance measure on the map? (c) The area of Geneva inside the city limits is approximately 16 km². How much area would this occupy on the 1 : 50 000 map (in cm²)?
(a) 6 000 m = 6 km (b) 24 cm (c) 64 cm²
**Doll-house furniture.** A doll house is 1 : 12. A real chair is 90 cm tall. (a) Find the doll-house chair height in cm. (b) A real living-room rug is 2.4 m × 1.8 m. Find the dimensions of the doll-house version. (c) The real chair weighs 8 kg. Assuming the doll-house chair is made of similar material at the same density, estimate its mass.
(a) 7.5 cm (b) 20 cm × 15 cm (c) ≈ 4.6 g
**Two similar boxes.** Two cuboid boxes are similar. The smaller has dimensions $a \times b \times c$ and volume $V$. The larger has linear scale factor $k$ relative to the smaller. (a) Express the larger volume in terms of $V$ and $k$. (b) The smaller box has volume 250 cm³. The larger box has volume 2000 cm³. Find $k$. (c) The smaller has surface area 200 cm². Find the surface area of the larger.
(a) $k^3 V$ (b) $k = 2$ (c) 800 cm²
**Building's shadow.** The Manhattan skyscraper "One World Trade Center" is 541 m tall. At noon, its shadow is 380 m long. A pedestrian 1.7 m tall is nearby. (a) Find the angle of elevation of the sun to 1 d.p. (b) Find the pedestrian's shadow length at the same time (assume parallel sunlight). (c) Demonstrate that the shadow triangles are similar.
(a) ≈ 54.9° (b) ≈ 1.19 m (c) See working
**Cake tin scaling.** A round cake tin has diameter 20 cm and depth 5 cm. A similar (linearly enlarged) cake tin is made with diameter 30 cm. (a) Find the linear scale factor. (b) Find the depth of the new tin. (c) The volume of cake batter required to fill each tin is proportional to its volume. The small tin needs 1.5 litres. Find the volume for the larger tin.
(a) 1.5 (b) 7.5 cm (c) ≈ 5.06 L (= 1.5 × 3.375)