Solved Problems: Scale Factor for Enlargements and Reductions

📚 Understanding Scale Factor

In geometry, the scale factor is the ratio between corresponding measurements of an object and a representation of that object. It's used to determine how much larger or smaller an image is compared to its original.

📜 History of Scale Factor

The concept of scaling has been used for centuries in mapmaking, architecture, and art. Early mathematicians and engineers recognized the importance of maintaining proportions when creating models or representations of real-world objects. The formalization of scale factor as a mathematical concept developed alongside the growth of geometry and cartography.

📐 Key Principles of Scale Factor

• 📏 Definition: The scale factor ($k$) is the ratio of a length on the image to the corresponding length on the original object.
• ➕ Enlargement: If $k > 1$, the image is an enlargement of the original.
• ➖ Reduction: If $0 < k < 1$, the image is a reduction of the original.
• 🧮 Calculation: To find the scale factor, divide the length of the image by the length of the corresponding side on the original object: $k = \frac{\text{Image Length}}{\text{Original Length}}$.
• ↔️ Units: Ensure that both lengths are measured in the same units before calculating the scale factor.

🌍 Real-World Examples

• 🗺️ Maps: Maps use scale factors to represent large geographical areas on a smaller surface. For example, a scale of 1:100,000 means that 1 unit on the map represents 100,000 units in reality.
• 🏗️ Architectural Models: Architects use scale models to visualize buildings before construction. A common scale is 1:50 or 1:100.
• 📷 Photography: When printing a photograph at a different size, the scale factor determines how much the image is enlarged or reduced.
• 🔬 Microscopy: Microscopes use scale factors to enlarge tiny objects, making them visible to the naked eye.

✍️ Solved Problems

Problem 1: Enlargement

A rectangle has a length of 5 cm and a width of 3 cm. It is enlarged by a scale factor of 2. What are the dimensions of the enlarged rectangle?

Solution:

• 📏 New Length = Original Length $\times$ Scale Factor = $5 \text{ cm} \times 2 = 10 \text{ cm}$
• 📐 New Width = Original Width $\times$ Scale Factor = $3 \text{ cm} \times 2 = 6 \text{ cm}$

Problem 2: Reduction

A square has a side length of 8 inches. It is reduced by a scale factor of 0.5. What is the side length of the reduced square?

Solution:

• 📏 New Side Length = Original Side Length $\times$ Scale Factor = $8 \text{ inches} \times 0.5 = 4 \text{ inches}$

Problem 3: Finding the Scale Factor

An object is 4 cm tall in a photograph and 20 cm tall in real life. What is the scale factor?

Solution:

• ➗ Scale Factor = $\frac{\text{Image Height}}{\text{Original Height}} = \frac{4 \text{ cm}}{20 \text{ cm}} = 0.2$

📝 Practice Quiz

1. A triangle with a base of 6 cm is enlarged by a scale factor of 3. What is the base of the enlarged triangle?
2. A line segment of 10 inches is reduced by a scale factor of 0.4. What is the length of the reduced line segment?
3. A circle has a radius of 2 cm. It is enlarged by a scale factor of 2.5. What is the radius of the enlarged circle?
4. A square with a side length of 12 cm is reduced to a side length of 3 cm. What is the scale factor?
5. A rectangle with a length of 8 cm and a width of 5 cm is enlarged by a scale factor of 1.5. What are the dimensions of the enlarged rectangle?

💡 Tips and Tricks

• ✅ Always double-check that your units are consistent before calculating the scale factor.
• ✅ Remember that a scale factor greater than 1 indicates an enlargement, while a scale factor between 0 and 1 indicates a reduction.
• ✅ Practice with various examples to build your understanding and confidence.

🔑 Conclusion

Understanding scale factors is crucial in various fields, from mapmaking to architecture. By grasping the basic principles and practicing with real-world examples, you can master this essential concept. Keep practicing, and you'll find working with scale factors becomes second nature!