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Enlargement is a mathematical concept that is used to describe the process of increasing the size of a shape or object, without altering its shape or proportions. It is commonly used in geometry, and it is a fundamental concept that is essential for various applications in science and engineering.

In this article, we will explore the concept of enlargement, including its definition, properties, and applications. We will also provide several examples to help you understand the concept better.

Definition of Enlargement

Enlargement is a transformation that changes the size of a shape, but not its shape or orientation. It is a type of dilation, which is a transformation that changes the size of a shape by a constant factor, without altering its shape or proportions.

The factor by which a shape is enlarged is called the scale factor. The scale factor is a ratio that compares the size of the original shape to the size of the enlarged shape. It is usually represented as a fraction, such as 2/1, 3/1, or 4/1, where the numerator represents the length of the enlarged shape, and the denominator represents the length of the original shape.

Properties of Enlargement

Enlargement has several important properties that are useful in geometry and other mathematical applications. Some of these properties include:

1. Enlargement preserves shape: When a shape is enlarged, its shape is preserved. This means that the shape remains the same, even though its size has changed.
2. Enlargement preserves orientation: Enlargement also preserves the orientation of a shape. This means that the shape remains in the same position and orientation, even though its size has changed.
3. Enlargement is a type of dilation: Enlargement is a type of dilation, which means that it changes the size of a shape by a constant factor, without altering its shape or proportions.
4. Enlargement has a scale factor: The factor by which a shape is enlarged is called the scale factor. The scale factor is a ratio that compares the size of the original shape to the size of the enlarged shape.
5. Enlargement is a linear transformation: Enlargement is a linear transformation, which means that it can be represented by a matrix.

Examples of Enlargement

Here are ten examples of enlargement that illustrate the concept:

• Enlarge a square by a scale factor of 2:

First, draw a square with sides of length 2 units. To enlarge the square by a scale factor of 2, multiply the length of each side by 2 to get a new square with sides of length 4 units.

• Enlarge a triangle by a scale factor of 3:

First, draw a triangle with sides of length 2 units, 3 units, and 4 units. To enlarge the triangle by a scale factor of 3, multiply the length of each side by 3 to get a new triangle with sides of length 6 units, 9 units, and 12 units.

• Enlarge a rectangle by a scale factor of 4:

First, draw a rectangle with sides of length 3 units and 4 units. To enlarge the rectangle by a scale factor of 4, multiply the length of each side by 4 to get a new rectangle with sides of length 12 units and 16 units.

• Enlarge a circle by a scale factor of 2:

First, draw a circle with a radius of 2 units. To enlarge the circle by a scale factor of 2, multiply the radius by 2 to get a new circle with a radius of 4 units.

• Enlarge a hexagon by a scale factor of 3:

First, draw a regular hexagon with sides of length 2 units. To enlarge the hexagon by a scale factor of 3, multiply the length of each side by 3 to get a new hexagon with sides of length 6 units.

• Enlarge a parallelogram by a scale factor of 1.5:

First, draw a parallelogram with sides of length 4 units and 6 units. To enlarge the parallelogram by a scale factor of 1.5, multiply the length of each side by 1.5 to get a new parallelogram with sides of length 6 units and 9 units.

• Enlarge a pentagon by a scale factor of 2:

First, draw a regular pentagon with sides of length 3 units. To enlarge the pentagon by a scale factor of 2, multiply the length of each side by 2 to get a new pentagon with sides of length 6 units.

• Enlarge a trapezium by a scale factor of 1.2:

First, draw a trapezium with sides of length 5 units and 7 units, and a height of 4 units. To enlarge the trapezium by a scale factor of 1.2, multiply the length of each side by 1.2, and the height by 1.2 to get a new trapezium with sides of length 6 units and 8.4 units, and a height of 4.8 units.

• Enlarge a regular octagon by a scale factor of 3:

First, draw a regular octagon with sides of length 2 units. To enlarge the octagon by a scale factor of 3, multiply the length of each side by 3 to get a new octagon with sides of length 6 units.

• Enlarge a regular dodecagon by a scale factor of 4:

First, draw a regular dodecagon with sides of length 2 units. To enlarge the dodecagon by a scale factor of 4, multiply the length of each side by 4 to get a new dodecagon with sides of length 8 units.

FAQs about Enlargement

Q: What is the difference between enlargement and magnification? A: Enlargement and magnification are often used interchangeably, but they are not the same thing. Enlargement refers to the process of increasing the size of a shape or object, while magnification refers to the process of making an image appear larger.

Q: What is the formula for calculating the scale factor of an enlargement? A: The scale factor of an enlargement can be calculated by dividing the length of the enlarged shape by the length of the original shape. For example, if the length of the original shape is 2 units, and the length of the enlarged shape is 4 units, the scale factor would be 4/2, which simplifies to 2.

Q: How is enlargement used in real life? A: Enlargement is used in various real-life applications, such as in architecture, engineering, and design. For example, an architect may use enlargement to create a scaled-up model of a building design, while an engineer may use enlargement to design and test prototypes of machines and equipment.

Q: Can an object be enlarged and reduced at the same time? A: No, an object cannot be enlarged and reduced at the same time. Enlargement and reduction are opposite transformations, and they cancel each other out. If an object is enlarged by a scale factor of 2 and then reduced by a scale factor of 2, it will return to its original size.

Q: What is the difference between an enlargement and a transformation? A: An enlargement is a transformation that involves changing the size of a shape while keeping its shape and proportions the same. A transformation, on the other hand, refers to any change made to a shape, including changes in size, shape, orientation, or position.

Q: Is enlargement only used for 2D shapes? A: No, enlargement can be used for both 2D and 3D shapes. When enlarging a 3D shape, all dimensions (length, width, and height) must be increased by the same scale factor to maintain the same shape and proportions.

Q: What is the difference between an enlargement and a dilation? A: Enlargement and dilation are often used interchangeably, but they are slightly different. Enlargement refers to the process of increasing the size of a shape while maintaining its shape and proportions. Dilation, on the other hand, refers to the process of stretching or compressing a shape without necessarily maintaining its shape or proportions.

Q: Can an enlargement have a negative scale factor? A: Yes, an enlargement can have a negative scale factor. A negative scale factor indicates that the new shape is a reflection of the original shape, flipped across a line or plane.

Q: Can a shape be enlarged without using a scale factor? A: No, an enlargement must always use a scale factor to determine the new size of the shape. Without a scale factor, it would not be possible to know how much to increase the size of the shape.

Q: Is enlargement the same as stretching? A: Enlargement and stretching are similar, but not exactly the same. Enlargement refers to the process of increasing the size of a shape while maintaining its shape and proportions. Stretching, on the other hand, refers to the process of increasing the size of a shape in one direction, while compressing it in another direction, which changes the shape and proportions of the shape.

Quiz:

1. What is an enlargement?
2. How is the scale factor of an enlargement calculated?
3. What is the difference between an enlargement and magnification?
4. Can an object be enlarged and reduced at the same time?
5. What is the difference between an enlargement and a transformation?
6. Can enlargement be used for 3D shapes?
7. What is the difference between an enlargement and a dilation?
8. Can an enlargement have a negative scale factor?
9. Can a shape be enlarged without using a scale factor?
10. Is enlargement the same as stretching?

Conclusion:

Enlargement is a useful mathematical concept that allows us to increase the size of a shape while maintaining its shape and proportions. It is used in a variety of real-life applications, including architecture, engineering, and design. By understanding the principles of enlargement and practicing with examples, we can become more proficient in using this concept in our daily lives.