How to Do Scale Factor Dilation?

To perform a scale factor dilation, you need a center point of dilation and a scale factor. Here's how to do it:

1. Identify the Center of Dilation:

• The center of dilation is the fixed point from which the dilation occurs. All points on the original figure will either move towards or away from this point. The problem should specify the center of dilation; if not, assume it's the origin (0,0).

2. Determine the Scale Factor:

• The scale factor determines how much larger or smaller the image will be compared to the original figure (pre-image).
  • A scale factor greater than 1 indicates an enlargement.
  • A scale factor between 0 and 1 indicates a reduction.
  • A scale factor of 1 results in no change in size.
  • A negative scale factor indicates a dilation followed by a 180-degree rotation about the center of dilation.

3. Apply the Scale Factor to Each Point:

• For each point (x, y) on the original figure, multiply both the x-coordinate and the y-coordinate by the scale factor (k) to find the corresponding point on the dilated image. The new point will be (kx, ky).
• If the center of dilation is the origin (0,0): The calculation is straightforward:
  • Original Point: (x, y)
  • Dilated Point: (kx, ky)
• If the center of dilation is NOT the origin (h, k):
  1. Subtract the coordinates of the center of dilation (h, k) from the coordinates of the original point (x, y): (x-h, y-k).
  2. Multiply the result by the scale factor (s): (s(x-h), s(y-k)).
  3. Add the coordinates of the center of dilation (h, k) back to the result: (s(x-h) + h, s(y-k) + k).

4. Connect the New Points:

• Connect the dilated points in the same order as the original points to form the dilated image.

Example:

Let's say you have a triangle with vertices A(1, 2), B(3, 1), and C(2, 4), and you want to dilate it with a scale factor of 2, using the origin (0,0) as the center of dilation.

• A'(2*1, 2*2) = A'(2, 4)
• B'(2*3, 2*1) = B'(6, 2)
• C'(2*2, 2*4) = C'(4, 8)

Connect A', B', and C' to form the dilated triangle. This triangle will be twice the size of the original triangle.

Finding the Scale Factor (Reverse Process):

If you have the original figure and the dilated image, you can find the scale factor by:

1. Identifying corresponding points on the original figure and the dilated image.
2. Finding the center of dilation.
3. Measuring the distance from the center of dilation to a point on the original figure and the distance from the center of dilation to the corresponding point on the dilated image.
4. Dividing the distance from the center to the image point by the distance from the center to the pre-image point. This ratio is the scale factor.
   • Scale Factor = (Distance from center to image) / (Distance from center to pre-image)

Important Considerations:

• Dilation preserves the shape of the figure but changes its size. The angles remain the same.
• Lines that pass through the center of dilation remain unchanged during dilation.