Enlargement with Positive and Negative Scale Factors

Introduction

Key Concepts

Understanding Enlargement

Enlargement, also known as scaling, is a transformation that changes the size of a figure while maintaining its shape. This transformation involves a scale factor that determines how much the figure enlarges or reduces. Enlargement can be applied to all types of geometric figures, including points, lines, and shapes.

Scale Factors: Positive and Negative

The scale factor is a numerical value that dictates the degree of enlargement or reduction applied to a figure. It can be either positive or negative, each imparting distinct characteristics to the transformation.

• Positive Scale Factor: A positive scale factor increases the size of the figure without altering its orientation. For instance, a scale factor of 2 will double the dimensions of the original figure, preserving its shape and orientation.
• Negative Scale Factor: A negative scale factor not only changes the size but also reflects the figure across the origin, resulting in a mirror image. For example, a scale factor of -1.5 will increase the size by 50% and produce a reflection of the original figure.

Mathematical Representation

Enlargement can be represented mathematically using coordinate geometry. For a given point \( (x, y) \) in the original figure, the transformed point \( (x', y') \) after enlargement with a scale factor \( k \) is calculated as:

$$ x' = k \cdot x $$ $$ y' = k \cdot y $$

Where:

• \( k \) is the scale factor.
• Positive \( k \) results in an enlargement without reflection.
• Negative \( k \) results in an enlargement with reflection across the origin.

Properties of Enlargement

• Shape Preservation: Enlarged figures retain the same shape as the original.
• Proportional Dimensions: All linear dimensions of the figure are multiplied by the scale factor.
• Parallelism: Corresponding sides of the figure remain parallel after enlargement.

Examples of Enlargement

Consider a triangle with vertices at \( A(2, 3) \), \( B(4, 7) \), and \( C(6, 3) \). Applying an enlargement with a positive scale factor of 2:

$$ A' = (2 \cdot 2, 2 \cdot 3) = (4, 6) $$ $$ B' = (2 \cdot 4, 2 \cdot 7) = (8, 14) $$ $$ C' = (2 \cdot 6, 2 \cdot 3) = (12, 6) $$

The new triangle \( A'B'C' \) is twice the size of the original, maintaining its shape and orientation.

Now, applying a negative scale factor of -1:

$$ A' = (-1 \cdot 2, -1 \cdot 3) = (-2, -3) $$ $$ B' = (-1 \cdot 4, -1 \cdot 7) = (-4, -7) $$ $$ C' = (-1 \cdot 6, -1 \cdot 3) = (-6, -3) $$

The triangle \( A'B'C' \) is congruent to the original but reflected across the origin.

Applications of Enlargement

• Engineering and Design: Enlargement is used to create scale models of structures and mechanical parts.
• Art and Architecture: Artists and architects utilize scaling to develop designs and projections.
• Map Making: Maps are scaled representations of geographical areas, allowing for easier navigation and planning.

Challenges in Understanding Enlargement

• Negative Scale Factors: Grasping the concept of reflection combined with scaling can be challenging for students.
• Maintaining Proportions: Ensuring all dimensions are scaled uniformly requires careful calculation.
• Complex Figures: Enlarging figures with curves or irregular shapes demands a solid understanding of geometric principles.

Theoretical Explanations

Enlargement is a type of similarity transformation, where the enlarged figure is similar to the original. Two figures are similar if they have the same shape but different sizes, maintaining proportional relationships between corresponding sides and angles.

For a transformation to be an enlargement, there must be a center of enlargement, a fixed point through which all lines of transformation pass. The scale factor determines the size change relative to this center.

Vectors and Enlargement

In the context of vectors, enlargement affects both the magnitude and direction of the vectors. A positive scale factor stretches the vector, while a negative scale factor reverses its direction and stretches it.

For a vector \( \vec{v} = \langle x, y \rangle \), the enlarged vector \( \vec{v}' \) is given by:

$$ \vec{v}' = k \cdot \vec{v} = \langle kx, ky \rangle $$

This operation is fundamental in various vector transformations and applications in physics and engineering.

Enlargement in Coordinate Geometry

When dealing with coordinate geometry, enlargement requires applying the scale factor to each coordinate of the figure. This process can be applied to various shapes, including polygons, circles, and more complex figures.

For example, enlarging a circle with center \( (h, k) \) and radius \( r \) by a scale factor \( k \) results in a new circle with the same center but radius \( kr \).

Inverse of Enlargement

The inverse of an enlargement with scale factor \( k \) is an enlargement with scale factor \( \frac{1}{k} \). This inverse operation reduces the size of the figure proportionally.

For example, if a figure is enlarged by a scale factor of 3, its inverse transformation would involve a scale factor of \( \frac{1}{3} \), thereby reducing its size.

Enlargement vs. Other Transformations

Unlike other transformations such as translation, rotation, or reflection, enlargement specifically alters the size of the figure while maintaining its shape (in the case of positive scale factors) or shape and orientation (in the case of negative scale factors).

Practical Example: Enlarging a Rectangle

Consider a rectangle with vertices at \( A(1, 2) \), \( B(1, 4) \), \( C(5, 4) \), and \( D(5, 2) \). Applying an enlargement with a scale factor of 2:

$$ A' = (2 \cdot 1, 2 \cdot 2) = (2, 4) $$ $$ B' = (2 \cdot 1, 2 \cdot 4) = (2, 8) $$ $$ C' = (2 \cdot 5, 2 \cdot 4) = (10, 8) $$ $$ D' = (2 \cdot 5, 2 \cdot 2) = (10, 4) $$

The enlarged rectangle \( A'B'C'D' \) has doubled dimensions, with all sides and angles preserved.

Applying a negative scale factor of -1:

$$ A' = (-1 \cdot 1, -1 \cdot 2) = (-1, -2) $$ $$ B' = (-1 \cdot 1, -1 \cdot 4) = (-1, -4) $$ $$ C' = (-1 \cdot 5, -1 \cdot 4) = (-5, -4) $$ $$ D' = (-1 \cdot 5, -1 \cdot 2) = (-5, -2) $$

The resulting rectangle \( A'B'C'D' \) is reflected across the origin and maintains the same dimensions as the original, but in the opposite direction.

Comparison Table

Summary and Key Takeaways