Calculate a point's dilation image from its coordinates and scale factor, or find the scale factor from an original point and its image.

Coordinates must use the same units. Blank center coordinates mean the origin. Negative factors place the image on the opposite ray; zero collapses it to the center.

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Scale Factor Dilation Formula

The following formula is used to calculate the scale factor dilation of an image or shape.

SF = X2 / X1 = Y2 / Y1 (for nonzero coordinate displacements from the center)
  • Where SF is the scale factor dilation
  • X2 and Y2 are the final point’s coordinate displacements from the center
  • X1 and Y1 are the original point’s coordinate displacements from the center

Divide corresponding nonzero coordinate displacements from the dilation center and require the same factor in both coordinates. A zero original displacement must have zero image displacement. If the original point is the center, the factor is not uniquely determined.

Scale Factor Dilation Definition

A dilation scales corresponding lengths by the absolute value of its scale factor k. Corresponding areas scale by k². For a chosen center C, a point P maps to P′ = C + k(P − C).

Scale Factor Dilation Example

How to calculate a scale factor dilation?

  1. First, determine the change in X and Y of the original image.

    Measure the change in X and Y of a single length of the original image.

  2. Next, determine the change in X and Y of the modified image.

    Measure the change in X and Y of the same length on the modified image.

  3. Finally, calculate the scale factor dilation.

    Calculate the total scale factor dilation using the equation above.

Example Problem:

The following is a step-by-step example problem for calculating a scale factor dilation. This example follows the how-to calculate instructions from directly above.

First, the change in x and y of the original image must be determined. Measuring these two values we find these to be 5 and 7 respectively.

Next, the change in x and y of the modified image must be measured. For this example, these are found to be 10 and 14 respectively.

Finally, we can calculate the scale factor dilation using the formula above:

SF = X2/X1 = Y2/Y1 (for nonzero coordinate displacements from the center)

SF = 10/5 = 14/7

SF = 2 = 2

FAQ

What is scale factor dilation?

A scale factor dilation is a rate at which an image or shape is enlarged or shrunk to produce a scaled version of the image.

Scale Factor Dilation Calculator screenshot