Math worksheet on scale drawing with a real-world application involving highway measurements.

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Name: Date: Period:

Topic: Ratios, Proportions, Similar Figures

#### 1. The length of the leg in triangle ABC is 3 inches. The length of the corresponding leg in triangle DEF is 6 inches. What is the scale factor of triangle ABC to triangle DEF?
- Answer: Scale factor = $ \frac{3}{6} = \frac{1}{2} $

> ✔ So, the scale factor of triangle ABC to triangle DEF is 1:2.

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#### 2. If two triangles are similar, and one side of triangle ABC is 8 cm, and the corresponding side in triangle DEF is 12 cm, what is the ratio of similarity?
- Answer: Ratio = $ \frac{8}{12} = \frac{2}{3} $

> ✔ So, the ratio of similarity is 2:3.

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#### 3. A model car is 10 inches long. The actual car is 15 feet long. What is the scale of the model to the actual car?
- Convert 15 feet to inches: $ 15 \times 12 = 180 $ inches.
- Scale = $ \frac{10}{180} = \frac{1}{18} $

> ✔ So, the scale is 1:18.

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#### 4. A map has a scale of 1 inch = 5 miles. If two cities are 3 inches apart on the map, how far apart are they in real life?
- Distance = $ 3 \times 5 = 15 $ miles

> ✔ Answer: 15 miles

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#### 5. Two similar rectangles have areas of 16 sq ft and 64 sq ft. What is the ratio of their perimeters?
- Area ratio = $ \frac{16}{64} = \frac{1}{4} $
- Since area scales by the square of the linear dimensions:
- Linear ratio (side lengths) = $ \sqrt{\frac{1}{4}} = \frac{1}{2} $
- Perimeter ratio = same as linear ratio = $ \frac{1}{2} $

> ✔ So, the ratio of their perimeters is 1:2

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#### 6. A rectangle has a length of 12 cm and width of 8 cm. A similar rectangle has a length of 18 cm. What is the width of the second rectangle?
- Set up proportion:
$$
\frac{12}{18} = \frac{8}{x}
$$
Cross-multiply:
$ 12x = 144 $ → $ x = 12 $

> ✔ Width = 12 cm

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#### 7. A tree casts a shadow of 20 feet. At the same time, a 6-foot man casts a shadow of 4 feet. How tall is the tree?
- Use similar triangles:
$$
\frac{6}{4} = \frac{x}{20}
$$
Cross-multiply:
$ 4x = 120 $ → $ x = 30 $

> ✔ Tree height = 30 feet

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#### 8. Two similar triangles have corresponding sides of 9 cm and 15 cm. What is the scale factor of the smaller to the larger triangle?
- Scale factor = $ \frac{9}{15} = \frac{3}{5} $

> ✔ So, scale factor is 3:5

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#### 9. A blueprint uses a scale of 1 inch = 4 feet. If a room is 10 feet wide, how wide is it on the blueprint?
- $ \frac{1}{4} = \frac{x}{10} $
- $ 4x = 10 $ → $ x = 2.5 $

> ✔ Width on blueprint = 2.5 inches

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#### 10. A rectangle has a perimeter of 20 cm and a length of 6 cm. A similar rectangle has a length of 9 cm. What is the perimeter of the second rectangle?
- First, find width of first rectangle:
- Perimeter = $ 2(l + w) = 20 $
- $ 2(6 + w) = 20 $ → $ 6 + w = 10 $ → $ w = 4 $
- So, original rectangle: 6 cm × 4 cm
- Scale factor: $ \frac{9}{6} = 1.5 $
- New perimeter = $ 20 \times 1.5 = 30 $ cm

> ✔ Perimeter of second rectangle = 30 cm

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#### 11. A model airplane is 12 inches long. The actual airplane is 96 feet long. What is the scale of the model to the actual airplane?
- Convert 96 feet to inches: $ 96 \times 12 = 1152 $
- Scale = $ \frac{12}{1152} = \frac{1}{96} $

> ✔ Scale = 1:96

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#### 12. A photo is enlarged from 4 inches by 6 inches to 8 inches by 12 inches. Is this an enlargement? By what factor?
- Both dimensions doubled: $ \frac{8}{4} = 2 $, $ \frac{12}{6} = 2 $
- Yes, it's an enlargement by a factor of 2

> ✔ Answer: Yes, by a factor of 2

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#### 13. A scale drawing has a scale of 1:100. If a building is 50 meters long in real life, how long is it on the drawing?
- $ \frac{1}{100} = \frac{x}{50} $
- $ x = \frac{50}{100} = 0.5 $ meters = 50 cm

> ✔ Length on drawing = 50 cm

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#### 14. Two similar triangles have a ratio of similarity of 3:4. If the area of the smaller triangle is 27 sq cm, what is the area of the larger triangle?
- Area ratio = $ \left(\frac{3}{4}\right)^2 = \frac{9}{16} $
- Let $ A $ be area of larger triangle:
$$
\frac{27}{A} = \frac{9}{16}
$$
Cross-multiply: $ 9A = 432 $ → $ A = 48 $

> ✔ Area of larger triangle = 48 sq cm

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#### 15. A rectangle has a length of 10 cm and width of 5 cm. A similar rectangle has a length of 15 cm. What is its width?
- $ \frac{10}{15} = \frac{5}{x} $
- $ 10x = 75 $ → $ x = 7.5 $

> ✔ Width = 7.5 cm

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Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | 1:2 |
| 2 | 2:3 |
| 3 | 1:18 |
| 4 | 15 miles |
| 5 | 1:2 |
| 6 | 12 cm |
| 7 | 30 feet |
| 8 | 3:5 |
| 9 | 2.5 inches |
| 10 | 30 cm |
| 11 | 1:96 |
| 12 | Yes, by a factor of 2 |
| 13 | 50 cm |
| 14 | 48 sq cm |
| 15 | 7.5 cm |

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