Transformation: Enlargement Formula

Enlargement, also known as dilation, is a geometric transformation that changes the size of a shape or object while preserving its shape. Imagine projecting a shadow of an object – the shadow is an enlargement (or reduction) of the original object. Understanding enlargement transformations is crucial in various fields, from art and design to computer graphics and engineering. This guide provides a clear and concise overview of enlargement formulas, empowering you to perform and analyze scaling transformations with ease.

Understanding Enlargements: Scaling Up or Down

An enlargement is defined by two key elements:

• Center of Enlargement: The fixed point from which the shape is scaled. All points of the shape move directly away from or towards this point.
• Scale Factor (k): A number that determines the amount of enlargement or reduction.
  • If k > 1, the shape is enlarged.
  • If 0 < k < 1, the shape is reduced.
  • If k = 1, the shape remains unchanged.

Essential Enlargement Formulas (Center at the Origin):

Here's the key enlargement formula for points in a 2D coordinate plane, with the center of enlargement at the origin (0,0):

• Enlargement: (x, y) → (kx, ky) Both the x and y coordinates are multiplied by the scale factor k.

Enlargement about an Arbitrary Point:

Enlarging about a point other than the origin is more complex. It typically involves three steps:

1. Translate the shape so that the center of enlargement coincides with the origin.
2. Perform the enlargement using the formula above.
3. Translate the shape back to its original position.

Illustrative Examples: Putting Formulas into Practice (Center at Origin)

Let's work through some examples to understand how to apply the enlargement formula:

• Example 1: Enlarge the point (2, 3) by a scale factor of 2.

  • The enlarged point is (2 * 2, 3 * 2) = (4, 6).

• Example 2: Reduce the point (6, 9) by a scale factor of 1/3.

  • The reduced point is (6 * 1/3, 9 * 1/3) = (2, 3).

• Example 3: A triangle has vertices at (1, 1), (4, 1), and (2, 4). Enlarge the triangle by a scale factor of 1.5.

  • (1, 1) → (1 * 1.5, 1 * 1.5) = (1.5, 1.5)
  • (4, 1) → (4 * 1.5, 1 * 1.5) = (6, 1.5)
  • (2, 4) → (2 * 1.5, 4 * 1.5) = (3, 6)
  • The enlarged triangle has vertices at (1.5, 1.5), (6, 1.5), and (3, 6).

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Applications of Enlargement Transformations:

Enlargement transformations have applications in various fields, including:

• Art and Design: Creating scaled drawings and designs.
• Computer Graphics: Zooming in and out of images and scaling objects.
• Engineering: Designing scaled models and prototypes.
• Mapmaking: Representing geographical areas at different scales.
• Photography: Enlarging or reducing images.

Conclusion: Mastering Enlargement Transformations

Understanding enlargement transformations is essential for working with geometric shapes and their scaling. This guide provides a valuable resource for learning and applying these fundamental formulas. By mastering these concepts, you'll be better equipped to tackle challenges in mathematics, science, design, engineering, and various other fields.

Call to Action:

Bookmark this page for quick reference and share it with others who might find it helpful. Practice applying these formulas to different points and shapes to reinforce your understanding. Visualizing the enlargement process will solidify your grasp of this essential geometric transformation. Remember to pay close attention to the scale factor and the center of enlargement for accurate results.