What is a Dilation in Geometry? How to Find the Scale Factor of a Dilated Figure?

Dilation in Geometry How to Find the Scale Factor of a Dilated Figure

The process of dilation entails scaling or changing a thing. It is a transformation that uses the specified scale factor to make the objects smaller or larger. The pre-image is the original figure, while the image is the new figure that emerges via dilatation.

A circle with a radius of 10 units, for instance, is reduced to a circle with a radius of 5 units. This technique is applied in art and craft, photography, and logo design, among other fields.

Two forms of dilation exist.

  1. Expansion – an increase in object size.
  2. Contraction – a decrease in item size.

Dilation in Geometry

In geometry, dilation is the process of modifying an object’s or shape’s size without altering its shape. The form could be a point, a section of a line, a polygon, etc.

It should be noted that while the shape can be altered in size, the proportions of its various dimensions and angles never change.

Scale Factor of Dilation

A scale factor is a numerical value that can be used to alter the size of any geometric figure or object in relation to its original size. 

It is the ratio of the original figures and dilated figure sizes.

Since each angle is congruent, every segment is proportional, the slope of every segment is preserved, and the perimeter of the reimage and the image have the same scale factor, we may use dilation scale factors to compress or enlarge a figure to the size we need.

The scale factor is denoted by r or k.

  • If the scale factor is greater than 1 (k > 1), the image is enlarged.
  • If the scale factor is less than 1 (0 k 1), the image is condensed.
  • If the scale factor is 1 (k = 1), the image doesn’t change.

How to Find the Scale Factor of a Dilation

There are two ways to find the scale factor of dilation. The first is a manual and slow method and the second is the fast and online method through the center of dilation calculator. It provides you the results within a seconds without any error.

Get a pair of corresponding sides to determine the scale factor of a dilatation. The sides that are in the same place in both the old and new figures are said to be corresponding.

The corresponding sides of the old and new figures’ lengths must match. The ratio of the new length to the previous length can be used to calculate the scale factor. The scaling factor, k, is the outcome.

 Formula to Find Scalar Factor

Scale factor = new figure / original figure

The length of TU in polygon STUV is 4 units, which is the original value. The new figure’s corresponding side measures 1 unit in length. The ratio of the new length, 1, to the previous length, 4, is used to calculate the factor.

As a result, the scaling factor would be the ratio of 1/4. It represents a reduction because k = 1/4. The fact that the new figure is a scaled-down reproduction of the previous model serves as a confirmation of this.

How to Find the Center of Dilation

The origin (0, 0) of a coordinate plane is frequently the Centre of dilatation, although it can also be any fixed point on the plane. This is the location at which all points, whether inside, outside, or on a specific figure, are either increased or shrunk.

By utilizing two corresponding sides to calculate the scale factor, you may identify the center of dilatation. For the purpose of locating the center of dilation, choose a pair of matching sites.

How much farther the new point will be from the Centre of dilatation than the original point is indicated by the scale factor. Depending on the scale factor, the changes in x and y should be increased or decreased.

Related: Also learn the concepts of variance and covariance.

Frequently Asked Questions

How can you Calculate a Dilation’s scale factor?

Finding the ratio of the coordinated side lengths, where the new length is the numerator and the previous length is the denominator, yields the scale factor. For an enlargement, the scale factor will be more than 1, and for a decrease, it will be less than 1 but still greater than 0.

Where does the Dilation’s Centre lie?

The place in which all points are enlarged or contracted is known as the centre of dilation, and it can be found within, outside, or even on a particular figure.

How is the Scale Factor Known?

The variable k indicates the scaling factor. Depending on this value, the dilatation will either be an expansion or a decrease.

How do you Locate the Dilation’s Centre rather than its Origin?

You must first establish the scale factor in order to locate the Centre of dilatation rather than the origin. Once you know the scale factor, you may use the given formula to calculate the coordinates for the Centre of dilation by substituting a pair of matching coordinates and the scale factor.


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