How to Find the Scale Factor of a Dilation?

Dilation is the process of enlarging or reducing the size of a geometric object without deforming it. In this post, you will learn more about the concept of dilation and how to find the scale factor.

How to Find the Scale Factor of a Dilation?

Dilation is possible using the scale factor, which helps to increase or decrease the size of the object. A scale factor is a number by which the size of any shape or geometric figure can be changed relative to its original size.

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Step by step guide to finding the scale factor of a dilation

Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure that is obtained after dilation is called the image and the main image is called the pre-image. Dilation can be of two types:

  • Expansion – When the size of an object is increased.
  • Contraction – When the size of an object is decreased

Note the figure below, which shows the dilation of the square. In this dilation, the size of the square increases but the shape remains constant.

Center of dilation

Dilation geometry has an important concept called the “center of expansion”. Dilation transforms the size of the figure which may increase or decrease. Resizing occurs from a point called the center of dilation. It is the center of dilation from which objects/shapes expand or contract.  In the figure shown below, the triangle is enlarged from the center of dilation which is marked as \(R\).

The scale factor

A scale factor is a number by which the size of any geometrical figure or shape can be changed relative to its original size. It is the ratio of the sizes of the original figure with the dilated figure.

The scale factor can be denoted by \(r\) or \(k\).

  • The image is enlarged if the scale factor is more than \(1 (k > 1)\).
  • The image is contracted if the scale factor is less than \(1 (0< k <1)\).
  • The image remains the same if the scale factor is \(1 (k = 1)\).

Note: The magnitude of the scale factor is considered and the scale factor cannot be zero.

Scale factor formula

The scale factor can increase the size of an object or decrease the size of an object. The basic formula to find the scale factor of a dilated figure is:

\(\color{blue}{Scale\:factor\:=\:Dimension\:of\:the\:new\:shape\:\div \:Dimension\:of\:the\:original\:shape}\)

This formula can be written in another way that helps to find the dimensions of the new shape:

\(\color{blue}{Dimensions\:of\:the\:original\:shape\:×\:Scale\:factor\:=\:Dimension\:of\:the\:new\:shape}\)

Dilation in geometry

Dilation in mathematics is a process of changing the size of an object or a shape without deforming it. The shape can be a point, a line segment, a polygon, etc. It should be noted that the shape can be enlarged or reduced, but the ratio of each dimension of the shape and angles remain the same.

In the above figure:

\(ΔPQR\) is dilated (enlarged) to \(ΔP’Q’R’\) and the angles are the same. The coordinates of the vertices \(ΔPQR\) have changed after expansion.

\(P(1,3)  → P’ (3,9)\)
\(Q(3,1)  → Q’ (9,3)\)
\(R(1,1)  → R’ (3,3)\)

How to calculate the scale factor in dilation?

The scale factor can be calculated when the original dimension and the modified dimension are given. Let’s find the scale factor of a triangle with original dimensions and reduced dimensions.  As seen above, after dilation, the coordinates of \(ΔPQR\) changed as follows:

\(P(1,3)  → P’ (3,9)\)
\(Q(3,1)  → Q’ (9,3)\)
\(R(1,1)  → R’ (3,3)\)

Let’s consider each vertex one by one:

Vertex \(P\): The \(x\)-coordinate \(1\) changed to \(3\) and the \(y\)-coordinate \(3\) changed to \(9\). This shows that both the coordinates of \(P’\) became thrice the coordinates of \(P\).

Vertex \(Q\): The \(x\)-coordinate \(3\) changed to \(9\) and the \(y\)-coordinate \(1\) changed to \(3\). This again shows that both the coordinates of \(Q’\) are three times the coordinates of \(Q\).

Vertex \(R\): The \(x\)-coordinate \(1\) changed to \(3\) and the \(y\)-coordinate \(1\) changed to \(3\). As we can see, both the coordinates of \(R’\) became three times the coordinates of \(R\).

Therefore, the scale factor in this dilation is \(3\). Every coordinate of \(ΔPQR\) is multiplied by the scale factor of \(3\) to obtain the magnified \(Δ P’Q’R’\). In other words, if the scale factor is \(k\), the coordinates \((x, y)\) become \((kx, ky)\).

\((x,y)  → (kx,ky)\)

We learned the scale factor formula. In this case, if we divide the coordinates of the new vertices by the coordinates of the original vertices, we can get the scale factor. For example, let’s take the dimensions of vertex \(P (1, 3)\) and  \(P’ (3, 9)\).

  • Take the \(x\)-coordinate of \(P’ = 3\) and the \(x\)-coordinate of \(P = 1\).
  • Substitute the values in the formula: \(3 ÷ 1 = 3\).
  • Now, take the \(y\)-coordinate of \(P’= 9\)  and the \(y\)-coordinate of \(P=3\).
  • Apply the same formula: \(9 ÷ 3 = 3\). Thus, we get the scale factor of \(3\) from both coordinates.

Finding the Scale Factor of a Dilation – Example 1:

Find the scale factor of the circle with the measurements given in the figure.

Solution:

The circle has increased in its size. Therefore, the scale factor is: \(= Dimension\:of\:the\:new\:shape\:\div \:Dimension\:of\:the\:original\:shape\) \(=Radius\:of\:the\:larger\:circle\:\div \:\:Radius\:of\:the\:smaller\:circle\)

\(= 6  ÷ 1 = 6\)                         
So, the scale factor is \(6\).   

Exercises for Finding the Scale Factor of a Dilation

  • The length of each side of the square is \(6\) units. If the scale factor is \(3\), what will be the length of each side of the expanded square?
  • Find the scale factor of the dilation.
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{18}\)
  2. \(\color{blue}{2}\)

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