Enter the original coordinates and scale factor into the calculator to determine the new coordinates after dilation.

Transform a point about the origin.

Center of dilation is the origin (0, 0). Use one consistent coordinate scale. Negative k also reverses direction; k = 0 collapses all points to the origin.

Dilation Rule Formula

The following formula calculates image coordinates for a dilation centered at the origin (0, 0). Apply it separately to every point, endpoint or vertex.

(x', y') = (k × x, k × y)

Variables:

  • (x’, y’) are the new coordinates after dilation
  • (x, y) are the original coordinates before dilation
  • k is the scale factor

Multiply each original coordinate by k. Distances scale by |k|: values above 1 enlarge and values between 0 and 1 reduce. A negative k also reverses direction through the origin. At k = 0 every point collapses to the origin, so original coordinates cannot be uniquely recovered. For a different center, subtract the center before scaling and add it back afterward.

What is a Dilation Rule?

A dilation rule scales a figure about a fixed center. A nonzero factor preserves shape, with distances multiplied by the factor’s absolute value. This calculator uses the origin as the center; a negative factor also turns directions through the origin. Zero scale is a degenerate collapse, so it does not preserve a nondegenerate figure’s shape.

How to Calculate Dilation Rule?

The following steps outline how to calculate the Dilation Rule.

  1. First, determine the original coordinates (x, y).
  2. Next, determine the scale factor (k).
  3. Next, use the formula (x’, y’) = (k * x, k * y) to calculate the new coordinates after dilation.
  4. Finally, calculate the Dilation Rule.
  5. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Original coordinates (x, y) = (3, 5)

Scale factor (k) = 2