Mathematics
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Fluency · Pack A

8.4 Scale and Similarity

Answer each question. Show working where needed.

Bronze

  1. A map has scale 1 cm : 200 m. Find the real distance for 4 cm on the map.

  2. A scale is 1 : 50. A real length of 25 m is represented by what length on the drawing?

  3. A scale model is 1 : 18. A model car is 25 cm long. Find the length of the real car in metres.

  4. Two figures are similar with linear scale factor 3. The smaller has length 7 cm. Find the length on the larger figure.

  5. Two similar triangles. The smaller has shortest side 6 cm; the larger has shortest side 18 cm. Find the linear scale factor.

  6. On a map, 2 cm represents 1 km. Find the real distance for 8 cm.

  7. Are these triangles definitely similar? They have angles 50°, 60°, 70° and 50°, 60°, 70°. Justify.

  8. A rectangle on a 1 : 50 plan measures 6 cm by 4 cm. Find the real dimensions in metres.

  9. A scale drawing uses 1 cm : 5 m. The real garden is 35 m wide. Find the drawing width.

  10. Two figures are similar. State two properties of similar figures (give a sentence each).

Silver

  1. On a map with scale 3 cm : 600 m, a rectangle measures 6 cm by 9 cm. What are its real dimensions in metres?

  2. 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.

  3. A drawing is at 1 : 25. Two parallel lines on the drawing are 4 cm apart. Find the real distance.

  4. 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.

  5. A photo is enlarged by a linear scale factor of 1.5. The original is 10 cm by 15 cm. Find the new dimensions.

  6. Convert the scale "1 cm : 5 m" to the form "1 : k".

  7. Convert the scale "1 : 50 000" to the form "1 cm : ? km".

  8. A model dinosaur is 1 : 25 of life-size. Its leg in the model is 8 cm. Find the real leg length in m.

  9. Two similar rectangles have lengths 4 cm and 6 cm. Their widths are 3 cm and $w$ cm. Find $w$.

  10. 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).

Gold

  1. Two triangles are similar. The smaller has sides 3 cm and 4 cm; the larger has corresponding sides 6 cm and $x$ cm. Find $x$.

  2. 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 $\text{m}^2$.

  3. Two similar shapes have areas in the ratio 9 : 25. Find the linear scale factor.

  4. Two similar rectangles have a linear scale factor of 2. The smaller has area 30 cm². Find the area of the larger rectangle.

  5. 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.

  6. 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?

  7. A drawing at 1 : 200 shows a circle of diameter 5 cm. Find (a) the real diameter; (b) the real area of the circle.

  8. 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.

  9. 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.

  10. 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.

Platinum

  1. 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.

  2. 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 $a^2 : b^2$.

  3. 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.

  4. A photograph is to be enlarged so the area is doubled. Find the linear scale factor required, exact and to 3 d.p.

  5. A 1 : 200 scale model of a building uses 12 m² of card. Find the surface area of the real building in m².

  6. Triangles ABC and DEF are similar with $A \leftrightarrow D$ etc. Given $AB = 6$, $AC = 8$, $DE = 9$. Find $DF$.

  7. A 1 : 1000 model city has buildings that take up 50 cm² of model area in total. Find the real area in km².

  8. 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.

  9. 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.

  10. 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.