How to find center of dilation

General
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how to find center of dilation

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How to find the center of dilation

Answer: Determining the center of dilation in geometric transformations involves identifying a specific point from which objects are scaled either up or down. Here’s a detailed guide on how to find the center of dilation.

Understanding Dilation

Dilation is a transformation that produces an image that is the same shape as the original but is a different size. Important aspects of dilation include:

  1. Scale Factor (( k )): Determines how much the image is enlarged or reduced.
  2. Center of Dilation: The fixed point in the plane about which all points are expanded or contracted.

Steps to Find the Center of Dilation

  1. Identify Original and Dilated Points

    • You start with a set of points in the original figure and their corresponding points in the dilated figure. Let’s say points ( A ) and ( A’ ) where ( A ) is the original and ( A’ ) is the dilated image of ( A ).

  2. Draw Lines Through Pairs of Corresponding Points

    • Draw lines connecting each original point to its image point. For example, draw a line from ( A ) to ( A’ ), another from ( B ) to ( B’ ), and so on.

  3. Find Intersection of Lines

    • The point where these lines intersect is the center of dilation. To do this:
      • Use a ruler and draw the lines carefully.
      • Ensure the lines are extended far enough to meet.

  4. Verify Consistency

    • Often, more than two pairs of points are available. To ensure accuracy, you can use multiple pairs and check that all lines intersect at the same point.

Geometric Verification Method

Example Problem:

Suppose points ( A(2, 3) ) and ( A’(4, 6) ) are given, and you need to determine the center of dilation.

  1. Using Coordinates Method:

    • Write the coordinates of the points ( A = (x_1, y_1) ) and ( A’ = (x_2, y_2) ).
    • The coordinates given are ( A = (2, 3) ) and ( A’ = (4, 6) ).

  2. Calculate Scale Factor (( k )):

    • If the scale factor ( k = 2 ), then:
      A' = (k \cdot x_1, k \cdot y_1) = (2 \cdot 2, 2 \cdot 3) = (4, 6)
    • You can verify that dilated coordinates match.

  3. Deducing Center of Dilation:

    • The center of dilation can be found using the formula:
      k = \frac{A' - O}{A - O}
      where ( O ) is the center of dilation. Here, solve for ( O ):
      O = \frac{kx_1 - x_2}{k-1}, \frac{ky_1 - y_2}{k-1}
    • Substitute values k=2, x_1=2, y_1=3, x_2=4, y_2=6 :
      O = ( \frac{2 \cdot 2 - 4}{2-1}, \frac{2 \cdot 3 - 6}{2-1} ) = (0,0)

Graphical/Drawing Technique

  1. Using Graphing Tools:
    • Plot points on a coordinate plane.
    • Draw lines connecting original points to their corresponding images.
    • The intersection point is the center of dilation.

Practice Problems

  1. Find the center of dilation if ( B(1, 5) ) is dilated to ( B’(3, 15) ).

Steps:
- ( B = (1,5), B’ = (3,15) )
- Calculate ( k ) and then use the earlier method to find the center.

Conclusion

Finding the center of dilation involves connecting multiple points from the original and dilated figures. Identifying the intersection of these connecting lines provides the center. This process can be conducted algebraically or graphically for accuracy.

By following these steps and practicing more problems, you can master the method for determining the center of dilation for any set of geometric figures!