Transformasi Geometri Bagian 4 - Dilatasi (Perkalian) Matematika Wajib Kelas 11

Introduction

In this tutorial, we will explore the concept of dilatation in geometry, specifically focusing on its application in the 11th-grade mathematics curriculum. Dilatation, or scaling, is a transformation that alters the size of a geometric figure while maintaining its shape. This guide will break down the steps to understand and apply dilatation effectively.

Step 1: Understanding Dilatation

• Definition: Dilatation is the transformation that enlarges or reduces a shape by a scale factor relative to a fixed point known as the center of dilation.
• Scale Factor:
  • A scale factor greater than 1 enlarges the shape.
  • A scale factor between 0 and 1 reduces the shape.
• Center of Dilation: This is the point from which the dilation occurs. It can be any point on the plane.

Step 2: Identifying Points and Scale Factor

• Choose a Shape: Select a geometric figure (e.g., triangle, rectangle).
• Determine Points: Identify the coordinates of the vertices. For example, for a triangle with vertices A(1, 2), B(3, 4), and C(5, 6).
• Select a Scale Factor: Decide on a scale factor (e.g., 2 for enlargement or 0.5 for reduction).

Step 3: Applying the Dilatation Formula

• Transformation Formula: The new coordinates (x', y') after dilatation can be calculated using:

  • x' = k * (x - x₀) + x₀
  • y' = k * (y - y₀) + y₀
  • Where (x₀, y₀) are the coordinates of the center of dilation, and k is the scale factor.

• Example Calculation:

  • Let’s say the center of dilation is O(0, 0) and the scale factor k = 2.
  • For point A(1, 2):
    • x' = 2 * (1 - 0) + 0 = 2
    • y' = 2 * (2 - 0) + 0 = 4
  • Thus, the new coordinates A' are (2, 4).

Step 4: Transforming All Points

• Repeat the Calculation: Apply the transformation formula to all points of the shape.
  • For B(3, 4) and C(5, 6):
    • B' = (6, 8)
    • C' = (10, 12)

Step 5: Plotting the New Shape

• Graph the Original and Transformed Shape:
  • Use graph paper or a graphing tool.
  • Plot the original points A, B, C, and the transformed points A', B', C'.
• Connect the Points: Draw lines between the points to visualize the dilated shape.

Common Pitfalls to Avoid

• Incorrect Scale Factor Usage: Ensure you apply the correct scale factor for enlargement or reduction.
• Center of Dilation Misplacement: Make sure the center is accurately identified; it significantly affects the outcome.
• Neglecting to Plot: Always graphically represent both the original and transformed shapes for better understanding.

Conclusion

Dilatation is a fundamental concept in geometry that allows you to manipulate the size of shapes while preserving their proportions. By following these steps, you can effectively perform dilatation on various geometric figures. For further practice, try applying different scale factors and centers of dilation to enhance your understanding.