2D Shapes 3D Shapes Angles Area Circles Cones Coordinate Geometry Cylinders Dilation Distance Formula Equations Of Circles Euclidean Geometry Formulas For Perimeter Geometry Lines Measurement Midpoint Formula Planes Points Polygons Prisms Pyramids Quadrilaterals Reflection Rotation Spheres Surface Area Tangent Transformations Translation Triangles Volume

Dilation in Mathematics

Resizing Shapes: Expansions and Reductions

A Dilation is a transformation that changes the size of a figure but not its shape. Unlike translation, rotation, and reflection, which are "rigid" transformations (isometric), dilation is a non-rigid transformation.

[Image of dilation geometry]

When you dilate a shape, it gets bigger (enlargement) or smaller (reduction). It is the mathematical concept behind zooming in on a map, projecting a movie onto a screen, or dilating the pupils of your eyes.

1. Key Components of Dilation

To perform a dilation, you need two pieces of information:

  • Center of Dilation: A fixed point from which the shape expands or contracts. In most basic math problems, this is the Origin (0,0).
  • Scale Factor (k): A number that tells you how much to resize the shape.

2. The Scale Factor (k)

[Image of dilation scale factor example]

The scale factor k determines the nature of the dilation:

  • If k > 1: It is an Enlargement (Expansion). The image grows larger than the original.
  • If 0 < k < 1: It is a Reduction. The image becomes smaller than the original.
  • If k = 1: The image stays exactly the same size (Identity transformation).

3. The Coordinate Rule

When the center of dilation is the Origin (0,0), calculating the new coordinates is simple multiplication. You multiply both the x and y coordinates by the scale factor k.

Rule: (x, y) → (kx, ky)

4. Examples

Example 1: Enlargement

Dilate the point P(2, 3) by a scale factor of k = 3.

x' = 2 × 3 = 6
y' = 3 × 3 = 9
New Point P' = (6, 9)

Example 2: Reduction

Dilate the point Q(8, -4) by a scale factor of k = 0.5 (or ½).

x' = 8 × 0.5 = 4
y' = -4 × 0.5 = -2
New Point Q' = (4, -2)

5. Properties of Dilation

Although the size changes, dilation preserves certain properties, making the new shape similar to the original:

  • Angle Measures: Angles stay exactly the same.
  • Parallelism: Parallel lines remain parallel.
  • Collinearity: Points on a line remain on a line.
  • Orientation: The shape does not rotate or flip.

However, Distance is NOT preserved. The length of the sides changes according to the scale factor.

Conclusion

Understanding Dilation is crucial for grasping the concept of similarity in geometry. Whether scaling up a model for construction or resizing an image on a computer screen, dilation provides the mathematical rules for changing size without distorting shape.