Solved problems: Determining enlargements and reductions from scale factors

📚 Understanding Scale Factors: Enlargements and Reductions

A scale factor is a number that describes how much a figure is enlarged or reduced. It's a ratio between corresponding measurements of an object and a representation of that object. Understanding scale factors is crucial in fields like architecture, engineering, and graphic design.

📜 A Brief History

The concept of scaling has been used since ancient times, primarily in mapmaking and construction. Early architects and engineers used scaled drawings to represent large structures in a manageable format. The formalization of scale factors as a mathematical concept developed alongside the development of geometry and proportion.

📌 Key Principles

• 📏 Definition: A scale factor is the ratio of a length on a representation to the corresponding length on the original object.
• ➕ Enlargement: If the scale factor is greater than 1, the figure is an enlargement.
• ➖ Reduction: If the scale factor is between 0 and 1, the figure is a reduction.
• 🔢 Calculation: Scale factor = (Dimension of new shape) / (Dimension of original shape)

🧮 Determining Enlargements and Reductions

To determine whether a transformation is an enlargement or a reduction, you need to calculate the scale factor. Here’s how:

1. Identify Corresponding Lengths: Find a length on the original figure and its corresponding length on the new figure.
2. Calculate the Scale Factor: Divide the new length by the original length.
3. Interpret the Scale Factor:
   • If the scale factor > 1, it’s an enlargement.
   • If the scale factor < 1, it’s a reduction.
   • If the scale factor = 1, it’s neither an enlargement nor a reduction; it’s a congruence.

📐 Real-World Examples

Example 1: Enlargement

A photograph is enlarged from 4 inches wide to 12 inches wide. What is the scale factor?

Scale Factor = (New Width) / (Original Width) = 12 / 4 = 3

Since the scale factor is 3 (which is > 1), the photograph has been enlarged.

Example 2: Reduction

A map reduces a distance of 100 miles to 2 inches on the map. What is the scale factor?

Scale Factor = (Map Distance) / (Actual Distance) = 2 / 100 = 0.02

Since the scale factor is 0.02 (which is < 1), the map is a reduction.

Example 3: Determining Dimensions

A rectangle with dimensions 5 cm by 10 cm is enlarged by a scale factor of 2. What are the new dimensions?

• New Length = Original Length * Scale Factor = 10 cm * 2 = 20 cm
• New Width = Original Width * Scale Factor = 5 cm * 2 = 10 cm

The new dimensions are 10 cm by 20 cm.

💡 Tips and Tricks

• 🔍 Always double-check which dimension is the 'new' and which is the 'original'.
• ✍️ Simplify fractions when calculating scale factors to make interpretation easier.
• 🧮 Be mindful of units. Ensure the units are the same before calculating the scale factor.

📝 Practice Quiz

1. A square with a side length of 3 cm is enlarged to a square with a side length of 9 cm. What is the scale factor?
2. A line segment of 20 inches is reduced to 5 inches. What is the scale factor?
3. A triangle with a base of 8 cm is enlarged by a scale factor of 1.5. What is the new base length?
4. A rectangle with length 12 m and width 4 m is reduced by a scale factor of 0.25. What are the new dimensions?
5. A circle with a radius of 5 inches is enlarged to a radius of 15 inches. What is the scale factor?
6. A map has a scale of 1 inch = 50 miles. Two cities are 3.5 inches apart on the map. What is the actual distance between the cities?
7. A model car is built with a scale factor of 1:24. If the actual car is 12 feet long, how long is the model car in inches?

✅ Solutions

1. Scale factor = 9/3 = 3
2. Scale factor = 5/20 = 0.25
3. New base length = 8 * 1.5 = 12 cm
4. New length = 12 * 0.25 = 3 m, New width = 4 * 0.25 = 1 m
5. Scale factor = 15/5 = 3
6. Actual distance = 3.5 * 50 = 175 miles
7. Model car length = (12 * 12) / 24 = 6 inches

🎯 Conclusion

Understanding scale factors is essential for various applications, from creating accurate maps to designing scaled models. By grasping the basic principles and practicing with real-world examples, you can easily determine enlargements and reductions. Keep practicing, and you'll master this concept in no time!