Transformasi (3) - Dilatasi, Perkalian, Rumus Dilatasi, Soal Dilatasi - Matematika SMP

Introduction

This tutorial focuses on the concepts of dilation in geometry, specifically designed for ninth-grade mathematics. It covers the key formulas, methods for calculating dilation, and practical examples to enhance understanding. By following these steps, you'll gain clarity on how dilation works and how to apply it in various geometrical contexts.

Step 1: Understanding the Dilation Formula

• The basic formula for dilation is given by:

  [ D(P) = k \cdot P ]

  where:

  • ( D(P) ) is the dilated point.
  • ( k ) is the scale factor.
  • ( P ) is the original point.

• The scale factor ( k ) determines how much larger or smaller the figure will become. A scale factor greater than 1 enlarges the figure, while a scale factor between 0 and 1 reduces it.

Step 2: Dilation Around a Center Point

• To perform dilation around a specific center point ( O(x_o, y_o) ):

  1. Identify the original point ( P(x, y) ).

  2. Calculate the new point ( D ) using the formula:

     [ D(x', y') = (x_o + k \cdot (x - x_o), y_o + k \cdot (y - y_o)) ]

• Example:

  • If the center point is ( O(2, 3) ), the original point ( P(1, 1) ), and the scale factor ( k = 2 ):

    [ D(2 + 2(1 - 2), 3 + 2(1 - 3)) = (2 - 2, 3 - 4) = (0, -1) ]

Step 3: Finding the Initial Points Before Dilation

• To find a point before dilation, you can rearrange the dilation formula:

  [ P(x, y) = \left( \frac{x' - x_o}{k} + x_o, \frac{y' - y_o}{k} + y_o \right) ]

• This helps to reverse the dilation process to determine the original coordinates.

Step 4: Determining the Scale Factor

• The scale factor can be determined by comparing the distances:

  1. Measure the distance from the center point to the original point.
  2. Measure the distance from the center point to the dilated point.
  3. The scale factor ( k ) is the ratio of these two distances.

• Example:

  • If the distance from the center ( O ) to ( P ) is 3 units and to ( D ) is 6 units, then:

  [ k = \frac{\text{Distance to } D}{\text{Distance to } P} = \frac{6}{3} = 2 ]

Step 5: Dilation of Points Not Centered on the Origin

• If the center of dilation is not at the origin, follow the same steps as before, ensuring to account for the coordinates of the center point in your calculations.

Step 6: Multiple Dilations

• To perform two successive dilations:

  1. First, apply the dilation using the first scale factor.
  2. Take the new point obtained and apply the second dilation using the second scale factor.

• Example:

  • First dilation with ( k_1 = 2 ) and second dilation with ( k_2 = 0.5 ):

  [ D_1 = k_1 \cdot P \quad \text{and then} \quad D_2 = k_2 \cdot D_1 ]

Step 7: Dilation of Quadrilaterals

• To dilate a quadrilateral:
  1. Apply the dilation formula to each vertex of the quadrilateral.
  2. Use the center point and scale factor as discussed.

Step 8: Area Relationship with Dilation

• Understand that the area of a shape dilated by a factor ( k ) changes by a factor of ( k^2 ). Therefore:

  [ \text{New Area} = k^2 \cdot \text{Original Area} ]

Conclusion

In this tutorial, we covered the fundamental concepts of dilation in geometry, including the formulas and methods for calculating dilated points. Remember to apply the dilation formula correctly and understand the impact of the scale factor on both dimensions and area. Practice with different shapes and scenarios to solidify your understanding of dilations in geometry.