Transformasi (2) - Refleksi, Pencerminan, Rumus Pencerminan - Matematika SMP

Introduction

This tutorial focuses on the topic of reflection, a type of geometric transformation, as discussed in the video "Transformasi (2) - Refleksi, Pencerminan, Rumus Pencerminan." You will learn about the definition of reflection, the formulas involved, and practical examples that will enhance your understanding of this mathematical concept, particularly relevant for middle school mathematics.

Step 1: Understanding Reflection and Its Formula

• Definition of Reflection: Reflection is a transformation that creates a mirror image of a shape across a specific line, known as the line of reflection.
• Formula for Reflection:
  • For a point (x, y), the reflection across the x-axis is given by (x, -y).
  • The reflection across the y-axis is given by (-x, y).
  • For reflection across a vertical line x = a, the formula is:
    • Reflected point = (2a - x, y).
  • For reflection across a horizontal line y = b, the formula is:
    • Reflected point = (x, 2b - y).

Step 2: Example of Reflection Through a Central Point

• Example: Reflect the point (3, 4) through the origin (0, 0).
  • Apply the formula for reflection through the origin:
    • New point = (-x, -y) = (-3, -4).
• Practical Tip: Always visualize the original and reflected points on a graph for clarity.

Step 3: Reflecting Points Across a Vertical Line

• Example: Reflect the point (1, 2) across the line x = 2.
  • Using the formula:
    • Reflected point = (2*2 - 1, 2) = (3, 2).
• Common Pitfall: Ensure you correctly identify the line of reflection before applying the formula.

Step 4: Finding Original Points Before Reflection

• Scenario: You have a reflected point and need to find the original point before reflection.
• Example: Given the reflected point (4, 5) and the line x = 3, find the original point.
  • Use the reverse formula:
    • Original point = (2*3 - 4, 5) = (2, 5).
• Tip: Work backwards carefully to avoid calculation errors.

Step 5: Reflection of Shapes

• Reflecting a Rectangle:
  • Identify the coordinates of all vertices.
  • Apply the reflection formula to each vertex.
• Example: For a rectangle with vertices (1, 1), (1, 3), (3, 1), (3, 3) reflected across the line x = 2, the new vertices will be:
  • (3, 1), (3, 3), (1, 1), (1, 3).
• Practical Advice: Sketch the shape before and after reflection for better visualization.

Step 6: Reflection of Triangles

• Process:
  • List the vertices of the triangle.
  • Apply the reflection formula to each vertex.
• Example: Reflect a triangle with vertices (2, 1), (3, 5), and (5, 2) across the line y = 4.
  • New vertices will be (2, 7), (3, 3), and (5, 6).
• Tip: Confirm the new shape maintains the same area and angle measures.

Step 7: Reflection of Lines

• Understanding Line Reflection: Reflecting a line involves reflecting points on the line and determining the new equation.
• Example: Reflect the line y = x across the line y = 2.
  • Identify points on the line, reflect them, and find the new line equation.
• Practical Tip: Use graphing tools to visualize the line before and after reflection.

Conclusion

In this tutorial, you have learned the fundamentals of geometric reflection, including definitions, formulas, and practical examples involving points, shapes, and lines. To further solidify your understanding, practice reflecting various shapes and points across different lines. With these skills, you will have a solid foundation in geometric transformations, which is crucial for higher-level mathematics.