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Middle Years

7.08 Scale and area

Worksheet
Scaled area
1

Find the value, x, for each of the following similar rectangles:

a
b
c
d
2

The corresponding sides of two similar triangles are 6 \text{ cm} and 12 \text{ cm}. The area of the smaller triangle is 18 \text{ cm}^2. Find the area of the larger triangle.

3

A square has a side length of 8 \text{ cm}. Another square is created by scaling it up by a factor of 7. Find the area of the larger square.

4

Two rectangles are similar, with matching sides in the ratio 6:7. The smaller rectangle has an area of 576 \text{ cm}^2 and its shortest side has length 8 \text{ cm}. Find the length of the longest side of the bigger rectangle.

5

A triangle has side lengths of 6 \text{ cm}, 12 \text{ cm}, and 16 \text{ cm}. A second similar triangle has an area that is 9 times larger than the first.

a

Find the scale factor between the two triangles.

b

Describe the relationship between the perimeters of the two triangles.

6

A square of side length 6 cm is enlarged using the scale factor 3. Find the area of the enlarged square.

7

If a square with an area of 25\,\text{m}^{2} is dilated by a factor of 0.4, what is the side length of the dilated square?

Applications
8

A rectangular sign has a side length of 1.5 \text{ m}. In a photograph, the corresponding side has a length of 25 \text{ cm}.

a

Find the scale factor from the photograph to the sign.

b

The area of the sign is 10\,800 \text{ cm}^2 in real life. Determine its area in the photograph.

9

A model of the Eiffel tower is made in the ratio 1:12\,960.

a

The height of the Eiffel tower is 324 \text { m}. Find the height of the model in centimetres.

b

The model requires 15 \text{ mL} of paint to cover its entire surface area. How much paint, in litres, would be required to cover the entire surface area of the real Eiffel tower?

10

Maria runs a business where she makes and sells soft toys. Her most popular item is a teddy bear that is 14 \text{ cm} long. For a special promotion, she decides to make a larger version of the bear that is 56 \text{ cm} long.

a

Find the scale factor between the two bears.

b

Maria uses 224 \text{ cm}^2 of felt to make the small bear. How much felt will she need to make the big bear?

11

Bianca is looking over a map of her local area and notices that the scale of the map is given as 1:100 in the map legend.

a

Find the actual distance (in centimetres) between two points which are drawn 12 cm apart on that map.

b

Find the distance in metres between the two points from part (a).

12

The scale of the map is 1:2000 and two points are drawn 19 cm apart on the map.

a

Find the actual distance between the two points in centimeters.

b

Find the distance between the points in metres.

13

Consider a map with a scale of 1:25\,000.

a

Find the actual distance between two points which are drawn 14 cm apart on the map.

b

Find the distance between the points in kilometres.

14

The scale on a map is 1:400\,000. How far apart on the map should two train stations be drawn if the actual distance between the stations is 100 km? Give your answer in centimetres.

15

The scale on a map of a garden is 1:2000. How far apart on the map should two fountains be drawn if the actual distance between the fountains is 100 metres? Give your answer in centimetres.

16

The Dominican Republic is shown on the following map. According to the scale given, estimate the actual area of the Dominican Republic.

17

Consider two similar ceilings: the first with dimensions 5 m by 4 m, and the second with dimensions 20 m by 16 m.

a

Find the length scale factor.

b

Find the area scale factor.

c

The smaller ceiling took 1.5 L of paint to cover it. How many litres of paint would be required to paint the second ceiling?

18

The figure shows the field on which a game of tag is being played during a Physical Education class. To make it more of a fitness challenge, the teacher dilates the boundaries of the field by a factor of 1.5.

What is the area of the new field?

19

The figures below are similar. By first finding the ratio of the sides, what is the value of the pronumeral?

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