How to find the center of dilation

General
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how to find the center of dilation

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How to Find the Center of Dilation

Answer: Finding the center of dilation involves a series of steps and can be accomplished using geometric concepts. Let’s break down the process:

Step-by-Step Process to Find the Center of Dilation

  1. Understanding Dilation:

    • Dilation is a transformation that resizes a figure by a scale factor relative to a fixed point called the center of dilation.
    • If the scale factor is greater than 1, the figure enlarges. If it is between 0 and 1, the figure shrinks.

  2. Identifying Key Points:

    • Consider the original figure and its dilated image.
    • Identify corresponding points in the original and the dilated figure. For example, if a point ( A ) in the original figure corresponds to a point ( A’ ) in the dilated figure, then ( A \rightarrow A’ ).

  3. Drawing Lines:

    • Draw lines connecting each pair of corresponding points, like ( A ) to ( A’ ), ( B ) to ( B’ ), etc.
    • These lines are crucial as they will intersect at the center of dilation.

  4. Finding Intersections:

    • The point where all these lines intersect is the center of dilation.
    • If the lines appear to intersect at a single point, that is the center.
    • If they do not intersect at a single point, check your corresponding points and lines.

Example:

Let’s walk through an example for clarity.

  1. Given Data:

    • Original figure vertices: ( A(1,2) ), ( B(3,4) ).
    • Dilated figure vertices: ( A’(2,4) ), ( B’(6,8) ).

  2. Corresponding Points:

    • ( A \rightarrow A’ ) means (1,2) to (2,4).
    • ( B \rightarrow B’ ) means (3,4) to (6,8).

  3. Drawing connecting lines:

    • Draw a line from ( (1,2) ) to ( (2,4) ).
    • Draw another line from ( (3,4) ) to ( (6,8) ).

  4. Find Intersection:

    • Extend these lines as necessary.
    • The lines will intersect at the center of dilation.

Verification:

Check if the distances from the center to the original points and the scaled points align with the scale factor. For our example, the scale factor appears to be 2 (since (2,4) = 2 \times (1,2) and (6,8) = 2 \times (3,4) ). The center should fit this transformation accurately.

Summary:

The steps above outline the fundamental technique of determining the center of dilation. Understanding the principles and precisely identifying the corresponding points are key to accurately finding the center.

If you have specific points or questions, feel free to ask, and I can provide further detailed assistance!